divisibility rules by dr j c saini
DESCRIPTION
We present here rules/tests for the divisibility of an arbitrary integer by another by transforming the given number into a smaller one.TRANSCRIPT
DIVISIBILITY RULES/TESTS
JAGDISH SAINI
ABSTRACT. In this article we present rules/tests for the divisibility of an arbitrary integer by another in base 8,9,10,11,12,13,14,15 and 16 number systems. The rules/tests reduce the given number into a smaller one whose divisibility or other wise implies the divisibility of the original number.
Mathematics Subject Classification Number. 11A05
INTRODUCTI0N
The divisibility tests (in decimal system) for certain integers such as 2, 3, 4, 8, 9 and 11 are well known.
To test the divisibility by 3, 9 and 11 we form the sum of the digits or alternate digits of the given
number till we get a single digit number. The divisibility by 2, 4 and 8 is restricted to the divisibility of the
number formed by the last 1, 2 or 3 digits only of the given number. In other words we reduce the given
number in to a smaller number, whose divisibility implies the divisibility of the given number.
The question is: Can we reduce an arbitrary number into a smaller one whose divisibility or
otherwise by an integer ensures the divisibility of the given number? The answer is in affirmative. In
fact, in any base b-number system, for a divisor d co-prime to the base, every number can be reduced,
at least, to a Td - digit number, where Td is the period of the representation of the reciprocal of the
divisor in the base b. In case the divisor is a prime number (primality is base insensitive) and Td is even,
the dividend can be further reduced to a Td/2 - digit number.
To test whether an arbitrary number is divisible by a given integer db (in the base/scale of b) where
1. d10 ≤ 99910 and co-prime to 1010,
2. d8 ≤ 1448 = 10010 ,
3 d9 ≤ 1219 = 10010 ,
4. d11 ≤ 9111 = 10010 ,
5. d12 ≤ 8412 = 10010 ,
6. d13 ≤ 7913 = 10010
7. d14≤ 7214 = 10010 ,
8. d15≤ 6A15=10010 , and
1
9. d16 ≤6416 = 10010,
we provide, in tables 2,3,4,5,6,7,8,9 and 10 respectively given in the end of this paper, rules/tests
denoted by Jg(d)= m and J(d)g= m, where m, d and g are integers such that 0 <g ≤Td , 0 < ІmІ <d.
We define these rules/tests as;
Jg(d) = m: Starting from the end parse s into g-digit groups. Delete and multiply the leftmost group by
m and add to the next group. Repeat and continue the process till s is reduced to a ≤ g-digit
number say t.
J(d)g= m: Starting from the end parse s into g-digit groups. Delete and multiply the rightmost group by
m and add to the truncated s. Repeat and continue the process till the reduced number say t is ≤ a g-
digit number or starts increasing. If s is divisible by d, t may start repeating.
Then in both the cases s is divisible by d if and only if t is so. In these rules/tests ( given in table 2,3,4,5,6,7,8,9 and 10 ) the multiplier i. e. m is generally a single digit
number and where ever it is 2 or 3 digit number, all the digits are equal or only one of the digits is
different from 0 and 1. By this arrangement multiplication can be performed easily.
In fact, for a full period n-digit divisor d there are 2(d – n) rules /tests of each type mentioned above in
which 0 < ІmІ ≤ (d – 1). As an example (in decimal system) for d = 7, a full-period prime number, there
are 24 such rules/ tests.
Table 1
J1(7) = 3,
J4(7) = 4,
J(7)1 = 5,
J(7)4 = 2,
J1(7) = - 4,
J4(7) = -3,
J(7)1 = - 2,
J(7)4 = -5,
J2(7) = 2,
J5(7) = 5,
J(7)2 = 4,
J(7)5 = 3,
J2(7) = - 5,
J5(7) = - 2,
J(7)2 = - 3,
J(7)5 = - 4,
J3(7) = 6,
J6(7) = 1,
J(7)3 = 6,
J(7)6 = 1,
J3(7) = - 1,
J6(7) = - 6,
J(7)3 = - 1
J(7)6 = - 6.
Remark 1. Jg(d) = 1 simply means starting from the end parse the given dividend in to g-digit groups
and add all the groups.
Remark 2. J(d)1 = m simply means starting from the end delete and multiply the first digit of the
dividend by m and add to the truncated dividend.
To explain these rules/tests, we consider ( in decimal system) the divisibility of
69 78642789 67430093 67430876 by 137.
2
As given in the following table 2, the relevant rules/tests for 137 are
J8(137) = 1, J4(137) = - 1, J(137)1 = - 41.
J8(137) = 1 →: Starting from the end, we parse 69 78642789 67430093 67430876 in to 8-digit groups,
form their sum and repeat the process.
69 78642789 67430093 67430876 → 69 +78642789+ 67430093+ 67430876 = 2 13503827,
2 13503827 → 2+13503827 = 13503829. J4(137) = - 1 →: We now divide 1350 3829 into 4-digit groups, multiply the first group by - 1 and add to
the next group.
1350 3829 →3829-1350 =2479. J(137)1 = - 41 →: Now we multiply the last digit 9 i.e. the rightmost 1-digit group of 2479 by -41 and add
to 247 and repeat the process.
247 9 → 247-369 = - 122,
-12 2 → -12+82 = 70,
7 0 → 7 + 0 = 7.
Thus we have reduced 69 78642789 67430093 67430863 to 7 which is not divisible by 137. So, the given number 69 78642789 67430093 67430863 is also not divisible by 137.
The following table provides some divisibility rules/tests for integers 3-999 and co-prime to 10
TABLE 2 ( BASE -10 )
Divisor Divisibility rule/test
d Jg(d) = m / J(d)g = m
-------------------------------------------------------------------------------------------------------------------------------3 J1(3) = 1
7 J6(7) = 1, J3(7) = - 1, J(7)1= - 2
9 J1(9) = 1
11 J2(11) = 1,
13 J6(13) = 1,
J1(11) = - 1
J3(13) = - 1, J(13)1= 4 3
17 J16(17) =1,
19 J18(19) = 1,
J8(17) = -1,
J9(19) = - 1,
J4(17) = 4,
J5(19) = 3,
J(17)1= - 5
J(19)1= 2
21 J6(21) = 1, J(21)2 = 4
23 J22(23) = 1, J11(23) = - 1, J(23)3= - 2, J(23)1= 7
27 J3(27) = 1
29 J28(29) = 1, J14(29) = - 1, J8(29) = - 4, J( 29)3 = - 2, J(29)1= 3
31 J15(31) = 1, J6(31) = 2, J(31)1= - 3
33 J2(33) = 1
37 J3(37) = 1,
39 J6(39) = 1,
41 J5(41) = 1,
43 J21(43) = 1.
J(37)1= - 11
J(39)1 = 4
J(41)1 = - 4
J9(43) = - 2, J(43)2= - 3
47 J46(47) = 1,
49 J42(49) = 1,
J23(47) = - 1,
J21(49) = - 1,
J7(47) = - 2,
J13(49) = 3,
J(47)2= 8
J2(49) = 2
51 J16(51) = 1,
53 J13(53) = 1,
57 J18(57) = 1,
J2(51) = - 2
J6(53) = - 4, J(53)1 = 16
J8(57) = - 2
59 J58(59) = 1, J29(59) = - 1, J16(59) = - 11, J(59)4= - 2, J(59)1= 6
61 J60(61) = 1, J30(61) = - 1, J(61)13 = 2, J4(61) = - 4, J(61)1= - 6
63 J6(63) = 1, J3(63) = - 8
67 J33(67) = 1, J14(67) = - 11, J9(67) = 9, J(67)2= - 2
69 J22(69) = 1, J4(69) = - 5
4
71 J35(71) = 1, J(71)6= 2, J(71)1 = - 7
73 J8(73) = 1,
77 J6(77) = 1,
J4(73) = - 1, J(73)1= 22
J3(77) = - 1
79 J13(79) = 1, J5(79) = - 14, J(79)4 = - 12, J(79)1= 8
81 J9(81) = 1, J(81)1 = - 8
83 J41(83) = 1, J(83)19 = - 2, J(83)5 = 11, J7(83) = - 6, J3(83) = 4,
87 J28(87) = 1, J9(87) = - 11, J4(87) = - 5
89 J44(89) = 1, J22(89) = - 1, J(89)8 = 2, J(89)1 = 9
91 J6(91) = 1, J3(91) = - 1
93 J15(93) = 1, J7(93) = - 11
97 J96(97) = 1, J48(97) = - 1, J(97)10 = 2, J2(97) = 3
99 J2(99) = 1
101 J4(101) = 1, J2(101)= - 1
103 J34(103) = 1, J17(103) = -1, J12(103) = 8, J2(103) = -3,
107 J53(107) = 1,
109 J108(109) = 1,
J21(107) = - 2,
J54(109) = - 1,
J7(107) = - 6,
J(109)15= 2,
J2(107) = - 7
J(109)6 = - 16, J(109)1= 11
111 J3(111) = 1
113 J112(113) = 1, J56(113) = - 1, J23(113) = - 3, J12(113) = 14, J(113)4 = - 2,
117 J6(117) = 1
119 J48(119) = 1, J26(119) = 2, J(119)5 = 3
121 J22(121) = 1, J11(121) = - 1, J(121)1 = - 12
123 J5(123) = 1
5
127 J42(127) = 1, J21(127) = - 1, J6(127) = 2,
129 J21(129) = 1, J9(129) = - 2, J(129)3 = 4
131 J130(131) = 1, J65(131) = - 1, J18(131) = - 2, J(131)4 = 3
133 J18(133) = 1, J9(133) = - 1, J(133)2 = 4
137 J8(137) = 1, J4(137) = - 1, J(137)1 = - 41
139 J46(139) = 1, J23(139) = - 1, J10(139) = 6, J4(139) = - 8
141 J46(141) = 1, J14(141) = 4, J7(141) = - 2,
143 J6(143) = 1, J3(143) = - 1
147 J42(147) = 1, J(147)19 = - 2, J9!147) = 13, J4(147) = 4
149 J148(149) = 1, J74(149) = -1, J(149)31 = 2, J12(149) = - 4, J5(149) = 21
151 J75(151) = 1, J35(151) = 2, J17(151) = 11, J(151)5 = 4, J(151)1 = - 15
153 J16(153) = 1, J9(153) = 109, J6(153) = - 8
157 J78(157) = 1, J39(157) = -1, J22(157) = -3, J(157)9 = 4, J(157)2= 11
159 J13(159) = 1, J7(159) = 13
161 J66(161) = 1, J33(161) = -1, J8(161) = 2, J(161)4 = 9
163 J81(163) = 1, J(163)19 = - 3, J(163)5= - 2
167 J166(167) = 1, J83(167) = - 1, J(167)22 = 3, J13(167) = - 7, J6(167) = 4, J3(167) = - 2
169 J78(169) = 1, J39(169) = - 1, J(169)11 = - 3, J7(169) = 101, J(169)3 = 12
171 J18(171) = 1, J6(171) = - 8
173 J43(173) = 1, J(173)15 = - 4, J11(173) = 14, J5(173) = 6
177 J58(177) = 1, J16(177) = - 11, J(177)4= - 2
179 J178(179) = 1, J89(179) = -1, J(179)16 = - 2, J7(179) = - 14
6
181 J180(181) = 1, J90(181) = -1, J43(181) = - 2, J21(181) = 6, J7(181) = 112, J(181)4 = - 4,
183 J60(183) = 1, J34(183) = 4, J15(183) = - 11, J8(183) = 16
187 J16(187) = 1, J7(187) = - 12, J6(187) = 111
189 J6(189) = 1, J(189)2 = - 17
191 J95(191) = 1, J(191)44 = 2, J(191)21 = 3, J10(191) = - 11, J7(191) = 4
193 J192(193) = 1, J96(193) = - 1, J(193)36 = 3, J(193)10 = 2
197 J98(197) = 1, J49(197) = - 1, J38(197) = 101, J12(197) = - 4, J(197)6= - 7
199 J99(199) = 1, J28(199) = - 3, J8(199) = 112, J(199)4 = 4, J(199)2= 2
201 J33(201) = 1,
203 J84(203) = 1,
207 J22(207) = 1,
J13(201) = 19,
J( 203)31 = - 2,
J13(207) = - 8,
J(201)2 = - 2
J(203)7 = 12
J5(207) = 19
209 J18(209) = 1, J9(209) = - 1, J(209)4= 111
211 J30(211) = 1, J15(211) = - 1, J5(211) = - 14
213 J35(213) = 1, J(213)12 = 4, J4(213) = - 11
217 J30(217) = 1, J7(217) = - 11, J(217)4 = - 12
219 J8(219) = 1,
221 J48(221 = 1,
J(219)1 = 22
J19(221) = - 3, J(221)4 = - 4
223 J222(223) = 1, J111(223) = - 1, J18(223) = 2, J7(223) = 11
227 J113(227) = 1, J(227)47 = - 2, J30(227) = 101, J19(227) = 4, J7(227) = - 31
229 J228(229) = 1, J114(229) = - 1, J27(229) = 2, J(229)4= 3
231 J6(231) = 1, J(231)1 = - 23
233 J232(233) = 1, J116(233) = - 1, J(233) 32= 2, J10(233) = - 15, J(233)7 = 11, J(233)2= 7
7
237 J13(237) = 1, J5(237) = - 14
239 J7(239) = 1, J(239)1 = 24
241 J30(241) = 1, J15(241) = - 1, J5(241) = - 15, J(241)1= - 24
243 J27(243) = 1, J10(243) = 91
247 J18(247) = 1, J9(247) = - 1, J3(247) = 12, J(247)3 = 103
249 J41(249) = 1, J(249)19 = - 2, J3(249) = 4
251 J50 (251) =1, J25 (251) = -1. J14 (251) = - 2, J6(251) = 16, J3(251) = - 4
253 J22(253) = 1, J11(253) = - 1, J(253)3 = 21
257 J256 (257) = 1, J128 (257) = -1, J(257) 48 = - 2, J32 (257) = - 4, J28(257) = - 11,
J6 (257) = 13
259 J6(259) = 1. J(259)1= 26
261 J28(261) = 1, J11(261) = 118, J(261)9 = - 8
263 J262 (263) = 1, J131 (263) = -1, J25 (263) = - 2, J11(263) = -31, J4 (263) = 6, J(263)1= 79
267 J44(267) = 1, J14(267) = - 2, J(267)2 = - 8
269 J268 (269) = 1, J134 (269) = - 1, J(269)25= - 2, J12(269) = - 11, J(269)5 = - 91, J(269)1= 27
271 J5(271) = 1,
273 J6(273) = 1,
J(271)1= - 27
J(273)2 = - 101
277 J69 (277) = 1, J(277) 12 = - 4, J5 (277) = 3, J(277)1 = - 83
279 J15(279) = 1, J(279)4 = 19
281 J28 (281) = 1, J14 (281) = - 1, J(281)4 = 109, J(281)1 = - 28
283 J141 (283) = 1, J54(283) = - 2, J17( 283) = - 12, J(283)14 = 11, J10 (283) = 13, J(283)3= 15
287 J30(287) = 1, J19(287) = - 4, J8(287) = 16, J(287)4 = 51
8
289 J272(289) = 1, J42(289) = 2, J19(289) = 14, J5(289) = 6
291 J96(291) = 1, J38(291) = - 2, J23(291) = - 41, J13(291) = 112, J(291)5 = - 14
293 J146 (293) = 1, J73(293) = -1, J47 (293) = 4, J6 (293) = -9, J(293) 1= 88
297 J6(297) = 1
299 J66(299) = 1, J33(299) = - 1, J10(299) = 16, J(299)2 = 3
301 J42(301) = 1, J27(301) = - 8, J9(301) = 41, J(301)5 = 31, J(301)2 = - 3
303 J4(303) = 1, J(309)1 = 91
307 J153 (307) = 1, J(307)69 = - 2, J15(307) = 4, J9 (307) = -3 , J6 (307) = 101, J(307)1 = - 92
309 J34(309) = 1, J22(309) = 13, J9(309) = - 14, J(309)5 = - 8
311 J155 (311) = 1, J(311)71 = 2, J(311)22 = 3, J(311)9 = 12, J13(311) = 4, J7(311) = 106,
J(311)1 = - 31
313 J312(313) = 1, J156(313)= -1, J(313)38 = 2, J(313) 8 = - 11, J4(313) = - 16
317 J79(317) = 1,
319 J28(319) = 1,
J(317)11 = - 4,
J19(319) = 21,
J(317)4 = 11,
J7(319) = - 12,
J(317)1 = -95
J(319)1 = 32
321 J53(321) = 1, J21(321) = - 2, J7(321) = - 113
323 J144(323) = 1, J111(323) = 12, J(323)46= - 2, J13(323) = - 6, J6(323) = - 8
327 J108(327) = 1, J39(327) = - 2, J19(327) = - 14, J7(327) = 13
329 J138(329) = 1, J(329)16 = 2, J5(329) = - 16
331 J110(331) = 1, J55(331) = -1, J9(331) = 12, J3(331) = 7, J(331) 1= - 33
333 J3(333) = 1
337 J336(337) = 1, J168(337) = - 1, J(337)32 = 2, J16(337) = 13, J11(337) = 15, J3(337) = -11,
J(337)2 = 91
9
339 J112(339) = 1, J58(339) = 13, J30(339) = 31, J(339)4 = - 2
341 J30(341) = 1, J11(341) = - 12, J(341)1= - 34
343 J294(343) = 1, J147(343) = - 1, J(343)40 = 2, J(343)29 = 3, J12(343) = 15, J(343)5 = - 11
347 J173(347) = 1, J(347)28 = -2, J13(347) =12, J(347)4 = 11
349 J116(349) = 1, J58(349) = -1, J31(349) = 11, J15(349) = - 6, J11(349) =101, J8(349) = -17
351 J6(351) = 1, J(351)1 = - 35
353 J32(353) = 1, J16(353) = -1, J11(353) = - 7, J5(353) = 101, J(353)5= 7
357 J48(357) = 1, J(357)11= - 11, J4(357) = 4
359 J179(359) = 1, J(359)78 = 2, J(359)12 = 3, J11(359) = 12, J(359)3= - 14
361 J342(361) = 1, J171(361) = - 1, J80(361) = - 2, J35(361) = 21, J10(361) = 9, J5(361) = 3
363 J22(363) = 1, J12(363)10 = - 131, J(363)1 = 109
367 J366(367) = 1, J183(367) = -1, J(367)44 = 2, J11(367) = 12, J(367) 2= -11
369 J5(369) = 1, J(369)1 = 37
371 J78(371) = 1, J31(371) = - 11, J19(371) = - 4, J11(371) = - 9, J(371)4 = - 131
373 J186(373) = 1, J93(373) = - 1, J20(373) = - 3, J6(373) = -13, J(373)1 = 112
377 J84(377) = 1, J45(377) = - 14, J31(377) = 101, J21(377) = 12, J13(377) = - 3, J8(377) = - 4
379 J378(379) = 1, J189(379) = -1, J73(379) = 2, J(379)16 = - 3, J10(379) = 104, J(379)I = 38
381 J42(381) = 1, J(381)15 = - 2, J12(381) = 4
383 J382(383) = 1, J191(383) = -1, J59(383) = - 2, J28(383) = 6, J6(383) = -13, J(383)3 = - 18
387 J21(387) = 1, J6(387) = - 8
389 J388(389) = 1, J194(389) = -1, J(389)89 = - 2, J16(389) = - 4, J11(389) = - 101,
J6(389) = - 119
10
391 J176(391) = 1, J74(391) = 2, J40(391) = 101, J12(391) = 13
393 J130(393) = 1, J64(393) = 13, J18(393) = - 2, J7(393) = 115, J(393)3 = - 101
397 J99(397) = 1, J59(397) = 11, J47(397)= - 3, J(397)7 =12, J(397)5 = 9, J(397)1= - 119
399 J18(399) = 1, J6(399) = 106, J(399)2 = 4
401 J200(401) = 1, J100(401)= - 1, J49(401) = 2, J(401)19 = 11 J12(401) = 14, J(401)2 = - 4
403 J30(403) = 1, J8(403) = - 17
407 J6(407) = 1
409 J204(409) = 1, J102(409) = - 1, J47(409) = 3, J15(409) = 51, J6(409) = - 5, J(409)1= 41
411 J8(411) = 1, J(411)1 = - 41
413 J174(413) = 1, J87(413) = - 1, J43(413) = 101, J15(413) = 111, J7(413) = 31
417 J46(417) = 1, J22(417) = - 14, J4(417) = - 8
419 J418(419) = 1, J209(419) = - 1, J60(419) = -11, J49(419) = 2, J15(419) = - 111,
J(419)5 = 104, J(419)1 = 42
421 J140(421) = 1, J70(421) = - 1, J15(421) = 6, J7(421) = - 13, J(421)3 = 8
423 J46(423) = 1,
427 J60(427) = 1,
J21(423) = - 8,
J17(427) = - 2,
J(423)5= 91
J(427)9 = - 8
429 J6(429) = 1, J(429)1 = 43
431 J215(431) = 1, J(431)70 = 2, J39(431) = 11, J25(431) = 12, J5(431) = 8, J(431)1= - 43
433 J432(433) = 1, J216(433) = -1, J42(433) = - 2, J16(433) = 3, J8(433) = -51, J4(433) = 41
437 J198(437) = 1, J99(437) = - 1, J40(437) = 101, J16(437) = 4, J8(437) = - 21, J(437)5 = - 6
439 J219(439) = 1, J(439 )69 = 2, J33(439) = -3, J12(439) = 8, J7(439) = 19, J(439)1= 44
441 J42(441) = 1, J27(441) = - 8, J(441)7 = 19
11
443 J221(443) = 1, J(443)21 = - 2, J(443)7= - 11
447 J148(447) = 1, J(447)49 = - 11, J43(447) = - 2, J24(447) = 16, J(447)6 = 22
449 J32(449) = 1, J16(449) = - 1, J(449)3 = - 22, J(449)1 = 45
451 J10(451) = 1, J451)1= - 45
453 J75(453) = 1, J42(453) = - 101, J(453)33 = - 101, J16(453) = - 14, J(453)5 = 4
457 J152(457) = 1, J76(457) = - 1, J30(457) = 2, J(457)16 = - 4, J(457)5 = 11
459 J48(459) = 1, J38(459) = - 8, J(459)10 = - 8, J(459)1 = 46
461 J460(461) = 1, J230(461) = -1, J(461) 61= - 2, J28(461) = - 9, J7(461) = - 12, J(461)1 = - 46
463 J154(463) = 1, J77(463) = - 1, J58(463) = 111, J28(463) = - 101, J13(463) = 106. J5(463) = - 8
467 J233(467) = 1, J(467)61 = - 2, J44(467) = 3, J20(467) = - 11, J8(467) = - 111, J(467)2 = -14
469 J66(469) = 1, J50(469) = - 12, J(469)16 = - 12, J7(469) = - 18
471 J78(471) = 1, J69(471) = 4, J37(471) = - 11, J15(471) = - 101, J(471)9 = 4
473 J42(473) = 1,
477 J13(477) = 1,
J37(473) = - 12, J(473)5 = - 12
J4(477) = - 17
479 J239(479) = 1, J(479)33 = 2, J23(479) = 9, J(479)10 = 18, J5(479) = - 111, J(479)1= 48
481 J6(481) = 1
483 J66(483) = 1, J41(483) = - 2, J(483)25 = - 2, J16( 483) = 4, J(483)6 = 211
487 J486(487) = 1, J243(487) = -1, J(487)44 = 2, J18(487) = -12, J11(487) = 17, J(487)2 = - 112 489 j81(489) = 1, J71(489) = 4, J(489)10 = 4, J(489)5 = - 2 491 J490(491) = 1, J245(491) = -1, J48(491) = 11, J13(491) = 2 , J(491)5 = - 3, J(491)1 = - 49 493 J112(493) = 1, J37(493) = - 11, J27(493) = 3, J(493)20 = - 4, J7(493) = - 12 497 J210(497) = 1 J134(497) = 2, J109(497) = - 11, J79(497) = 3, J(497)76 = 2, J14(497) = - 117
12
J8(497) = 121, J3(497) = 6 499 J498(499) = 1, J249(499) = - 1. J204(499) = - 3, J98( 499) = - 11, J(499)45 = 3, J27(499) = 13,
J6(499) = 4, J3(499) = 2 501 J166(501) = 1, J152(501) = 31, J43(501) = 13, J27(501) = -11, J6(501) = 4, J(501)2 = - 5
503 J502(503) = 1, J251(503) = -1, J69(503) = -2, J25(503) = - 16, J(503)6 = 14, J3(503) = - 6 507 J78(507) = 1, J46(507) = - 101, J15(507) = 103, J(507)8 = 211 509 J508(509) = 1, J254(509) = - 1, J67(509) = 2, J(509)6 = 11, J(509)1 = 51 511 J24(511) = 1 513 J18(513) = 1, J7(513) = 91 517 J46(517) = 1, J23(517) = - 1, J9(517) = - 12, J(517)4 = 111 519 J43(519) = 1, J13(519) = 16, J(519)8 = 22, J(519)5 = 202 521 J52(521) = 1, J26(521) = - 1, J10(521) = - 18, J6(521) = 201, J(521)1 = - 52 523 J261(523) = 1, J47(523) = - 2, J13(523) = 7, J5(523) = 107 527 J240(527) = 1, J55(527) = 5, J(527)54 = 2, J37(527) = - 11, J16(527) = 103, J4(527) = - 13 529 J506(529) = 1, J253(529) = - 1, J113(529) = 11, J74(529) = 2, J18(529) = - 14, J5(529) = 19, 531 J58(531) = 1, 533 J30(533) = 1,
J(531)12 = - 8, J14(533) = - 4,
J(531)5= -71 J(533)2 = 16
537 J178(537) = 1, J116(537) = - 11, J(537)62 = - 11, J(537)16 = - 2, J7(537) = - 14 539 J42(539) = 1, J21(539) = - 1, J8(539) = - 131 541 J540(541) = 1, J270(541) = - 1, J113(541) = 2, J30(541) = - 15, J14(541) = -9, J(541)6 = 7 543 J180(543) = 1, J56(543)= - 11, J43(543) = - 2, J15(543) = 7, J(543)8= 16 547 J91(547) = 1, J13(547) = - 3, J5(547) = - 101 549 J60(549) = 1, J19(549) = - 17, J(549)9 = - 8 13
551 J252(551) = 1, J(551)73 = 2, J(551)37 = 21, J21(551) = 12, J8(551) = 112, J(551)3 = 27 553 J78(553) = 1, J(553)15 = - 15, J12(553) = 8, J7(553) = 101, J(553)4 = - 12 557 J278(557) = 1, J139(557) = -1, J32(557) = - 4, J20(557) = - 9, J(557)5 = - 15 559 J42(559) = 1, J19(559) = - 3, J(559)8 = 81, J(559)4 = 9 561 J16(561) = 1, J9(561) = 109, J(561)3 = - 23 563 J281(563) = 1, J(563)109 = - 2, J116(563) = 3, J63(563) = 4, J29(563) = 11, J7(563) = - 6
J(563)4 = 21 567 J18(567) = 1, J11(567) = 19, J(567)2 = - 17 569 J284(569) = 1, J142(569) = -1, J63(569) = - 2, J(569)11 = 7, J(569)7 = 13, J(569)1= 57 571 J570(571) = 1, J285(571) = -1, J(571)130 = - 2, J94(571) = 11, J28(571) = - 3, J16(571) = 202,
J7(571) = 77, J(571)1 = - 57 573 J95(573) = 1, j10(573) = - 11, J7(573) = 4 577 J576(577) = 1, J288(577) = - 1, J78(577) = 11, J(577)28 = - 2, J7(577) = 13 579 J192(579) = 1, J86(579) = - 2, J(579)20 = 4, J15(579) = 13, J(579)8 = 7 581 J246(581) = 1, J67(581) = 3, J(581)19 = - 2, J15(581) = 111, J(581)8 = - 101, J(581)4 = - 222 583 J26(583) = 1, J(583)4 = - 131, 587 J293(587) = 1, J(587)64= - 2, J45(587) = 12, J21 (587) = - 102, J9(587) = 301, J4(587) = 21 589 J90(589) = 1, J26(589) = 112, J14(589) = - 3, J(589)7 = 14 591 J98(591) = 1, J61(591) = 4, J24(591) = 16, J(591)7 = 19 593 J592(593) = 1, J296(593) = -1, J132(593) = 2, J75(593) = 3,, J(593)32 = - 4, J12(593) = -113,
J(593)6 = - 91 597 J99(597) = 1, J64(597) = 13, J16(597) = 7, J(597)4 = 4 599 J299(599) = 1, J(599)17 = 2, J15(599) = 3, J(599)10 = - 11, J(599)2 = 6 601 J300(601) = 1, J150(601) = - 1, J(601)24 = 2, J12(601) = - 111, J8(601) = 211, J(601)2 = - 6
14
603 J33(603) = 1 J15(603) = 91, J(603)6 = - 8 607 J202 (607) = 1, J101(607) = -1, J33(607) = - 311, J12(607) = - 6 609 J84(609) = 1, J53(609) = - 2, J37(609) = - 11, J22(609) = 4, J6(609) = 22, J(609)9 = - 8 611 J138(611) = 1, J69(611) = - 1, J24(611) = 131, J(611)10 = 9, J(611)5 = - 3 613 J51(613) = 1, 617 J88(617) = 1,
J14(613) = - 3, J44(617) = -1,
J5(613) = 81 J14(617) = - 15, J(617)7 = -101
619 J618(619) = 1, J309(619) = -1, J(619)139 = 2 , J81(619) = 3, J31(619) = - 4, J13(619) = - 7
J(619)3 = 13 621 J66(621) = 1, J35(621) = - 8, J19(621) = - 71, J5(621) = 19 623 J132(623) = 1, J80(623) = 2, J28(623) = 4, J8(623) = 401 627 J18(627) = 1, J11(627) = 109, J(627)7 = 109 629 J48(629) = 1, J33(629) = 112, J16(629) = - 101, J10(629) = 121, J5(629) = - 11 631 J315(631) = 1, J(631)98 = 2 , J21(631) = 8 J13(631) = - 109 633 J30(633) = 1, 637 J42(637) = 1,
J11(633) = 61, J21(637) = - 1,
J5(633) = -14 J10(637) = 81, J5(637) = - 9
639 J35(639) = 1, J21(639) = - 17, J13(639) = 109, J(639)5 = 91 641 J32(641) = 1, J16(641) = - 1, J10(641) = 16, J5(641) = 4, J(641)1= - 64 643 J107(643) = 1, J19(643) = - 2, J8(643) = - 3 647 J646(647) = 1, J323(647) = - 1, J66(647) = 2, J47(647) = 15 , J15(647) = - 112, J(647)6 = 121,
J(647)3= 11, 649 J58(649) = 1, 651 J30(651) = 1,
J29(649) = - 1, J(649)5 = - 12 J7(651) = - 11
653 J326(653) = 1, J163(653) = -1, J87(653) = 11, J67(653) = 4, J39(653) = 13, J15(653) = 9,
J5(653) = 91
15
657 J8(657) = 1 659 J658(659) = 1, J329(659) = - 1, J149(659) = 2, J78(659) = 3, J47(659) = - 12, J13(659) = - 311
J(659)5= - 51, J(659)1= 66 661 J220(661) = 1, J110(661) = - 1, J40(661) = - 3, J17(661) = 8, J(661)5= 7, J(661)1= - 66 663 J48(663) = 1, J24(663) = 118, J(663)8 = 16 667 J308(667) = 1, J135(667) = 11, J97(667) = - 3, J73(667) = - 101, J19(667) = 21, J4(667) = -5 669 J222(669) = 1, J118(669) = - 11, J36(669) = 4, J11(669) = 61, J8(669) = - 113 671 J60(671) = 1, J45(671) = - 111, J16(671) = 12, J5(671) = 21, J(671)1 = - 67 673 J224(673) = 1, J112(673) = -1, J(673)24 = - 12, J10(673) = 7, J7(673) = - 107 677 J338(677) = 1, J169(677) = - 1, J(677) 66= - 4, J33(677) = -13, J22(677) = -117 J6(677) = 71 679 J96(679) = 1, J(679)10 = 2, J11(679) = 5, J(679)3 = 55 681 J113(681) = 1, J66(681) = - 2, J(681)47 = - 2, J32(681) = 112, J19(681) = 4, J5(681) = - 107
J(681)4 = 19 683 J341(683) = 1, J62(683) = -2, J(683)11 = 3, J5(683) = - 401 687 J228(687) = 1, J141(687) = - 2, J(687)87 = - 2, J(687)36 = - 11, J61(687) = 112 689 J78(689) = 1, J32(689) = - 4, J11(689) = - 9 691 J230(691) = 1, J115(691) = -1, J74(691) = 33, J9(691) = 2, J(691)1= - 69 693 J6(693) = 1 697 J80(697) = 1, J44(697) = - 4, J11(697) = - 31, J(697)5 = 411 699 J232(699) = 1, J84(699) = - 2, J42(699) = 109, J9(699) = 115, J(699)2 = 7 701 J700(701) = 1, J350(701) = - 1, J109(701) = - 2,, J34(701) = 31, J16(701) = 22 , J(701)10 = 17
J(701)2 = - 7 703 J18(703) = 1, 707 J12(707) = 1,
J(703)5 = - 101 J(707)3 = 111
16
709 J708(709) = 1, J354(709) = - 1, J137(709) = - 2, J104(709) = - 11, J15(709) = 13, J5(709) = 31,
J(709)1 = 71 711 J13(711) = 1, J(711)5 = - 17 713 J330(713) = 1, J96(713) = 2, J18(713) = 101, J(713)8 = - 11 717 J7(717) = 1 719 J359(719) = 1, J(719)160 = 2, J60(719) = 3, J39(719) = 4 J28(719) = 111, J24(719) = 31,
J10(719) = - 114, J(719)9 = - 101 721 J102 (721) = 1, J51(721) = - 1, J19(721) = 3, J12(721) = 8 723 J30(723) = 1, J19(723) = - 119, J6(723) = 91, J(723)5 = 16 727 J726(727) = 1, J363(727) = - 1, J63(727) = - 2, J21(727) = - 9, J(727)6= 411 729 J81(729) = 1, J40(729) = 118, J(729)7 = 19 731 J336(731) = 1, J114(731) = - 2, J45(731) = 11, J19(731) = - 3, J6(731) = - 8 733 J61(733) = 1, J36(733) = - 4, J11(733) = 16, J(723)2 = 22 737 J66( 737) = 1, J41(737) = 21, J43(737) = - 111, J17(737) = - 12, J6(737) = - 109 739 J246(739) = 1, J123(739) = - 1, J8(739) = - 2, J(739)1= 74 741 J18(741) = 1, J(741)3 = 103 743 J742(743) = 1, J371(743) = - 1, J(743)68 = 2, J19(743) = 21, J11(743) = 14, J(743)6 = - 181 747 J41(747) = 1, J25(747) = - 8, J10(747) = - 107, J(747)6 = 109 749 J318(749) = 1, J145(749) = 3, J(749)85 = - 2, J45(749) = 4 1, J7(749) = 101 751 J125(751) = 1, J38(751) = - 7, J24(751) = - 111, J10(751) = 171, J(751)1 = - 75 753 J50(753) = 1, J14(753) = -2, J6(753) = 16 757 J27(757) = 1, J6(757) = 3, J(757)3 = 81 759 J22(759) = 1 J(759)5 = 109 761 J580(761) = 1, J190(761) = - 1, J31(761) = 2, J19(761) = 41, J(761)12 = 401, J(761)3 = 121
17
763 J108(763) = 1, J82( 763) = - 3, J34(763) = 102, J9(763) = - 8 767 J174(767) = 1, J87(767) = - 1, J61(767) = - 3, J43(767) = 101, J13(767) = - 16. J(767)6 = 105 769 J192(769) = 1, J96(769) = - 1, J23(769) = - 4, J4(769) = 3 771 J256(771) = 1. J(771)48 = - 2, J28(771) = - 11, J6(771) = 13, J(771)3 = - 101 773 J193(773) = 1, J103(773) = - 4, J17(773) = -11, J(773)7 = - 113 777 J6(777) = 1 779 J90(779) = 1, 781 J70(781) = 1, 783 J84(783) = 1,
J(779)11 = - 4, J32(781) = 12, J47(783) = - 8,
J(779)3 = 141 J24(781) = 111, J16(783) = - 71,
J13(781) = 109 J6(783) = 109
787 J393(787) = 1, J173(787) = - 2, J56(787) = 6, J18(787) = - 211 J5(787) = 51 789 J262(789) = 1, J25(789) = - 2, J10(789) = 181 791 J336(791) = 1, J79(791) = 3, J22(791) = 11, J(791)8 = 4 793 J60(793) = 1, J24(793) = 131 797 J199(797) = 1, J(797)52 = 6, J37(797) = 111, J9(797) = 115, J7(797) = 41 799 J368(799) = 1, J122(799) = 2, J(799)19 = 11, J(799)10 = 9, J(799)2 = 8 801 J44(801) = 1, J19(801) = - 17, J(801)8 = 91, J(801)2 = - 8 803 J8(803) = 1 807 J268(807) = 1, J(807)25 = - 2, J12(807) = - 11 809 J202(809) = 1, J101(809) = - 1, J32(809) = - 4, J12(809) = 113, J(809)1 = 81 811 J810(811) = 1, J405(811) = -1, J101(811) = 3, J(811)78 = - 2, J23(811) = - 6, J(811)10 = - 119
J(811)1 = - 81 813 J5(813) = 1 817 J126(817) = 1, J65(817) = 14, J24(817) = 11, J6(817) = - 8 819 J6(819) = 1,
18
821 J820(821) = 1, J410(821) = -1, J197(821) = - 2, J157(821) = - 3, J141(821) = 12, J86(821) = 101
J26(821) = 7, J6(821) = 22 823 J822(823) = 1, J411(823) = - 1, J(823)122 = 2, J28(823) = - 7, J13(823) = - 201, J(823)2 = 107 827 J413(827) = 1, J176(827) = - 2, J39(827) = 3, J29(827) = - 7, J8(827) = - 13 829 j276(829) = 1, j138(829) = - 1, J102(829) = 12, j29(829) = 7, j17(829) = 18 J(829)6= - 11 831 J69(831) = 1, J48(831) = 211, J25(831) = -311, J19(831) = 202 833 J336(833) = 1, J37(833) = - 11, J8(833) = 16, J4(833) = 4 837 J15(837) = 1, J11(837) = 19, J(837)4 = 19 839 J419(839)= 1, J114(839) = 2, J14(839) = 3, J10(839) = 111, J7(839)= - 41 841 J812(841) = 1, J406(841) = - 1, J67(841) = 2, J20(841) = 7, J6(841) = 51 843 J28(843) = 1, J16(843) = 181, J8(843) = 811, J(843)4 = 109 847 J66(847) = 1, J33(847) = - 1, J23(847) = 131, J(847)10 = - 131 849 J141(849) = 1, J54(849) = - 2, J10(849) = 13, J(849)5 = - 14 851 J66(851) = 1, J14(851) = - 11, J7(851) = - 101 853 J213853) = 1, J53(853) = - 4, J31(853) = 14, J(853)6 = - 3 857 J856(857) = 1, J428(857) = - 1, J(857)166 = 2, J96(857) = - 4, J46(857) = 13, J25(857) = - 511.
J(857)11 = 105, J(857)3 = 6 859 J26(859) = 1, 861 J30(861) = 1,
J13(859) = -1, J(859)1= 86 J8(861) = 16
863 J862(863) = 1, J431(863) = - 1, J142(863) = 2, J69(863) = 201, J49(863) = - 6, J22(863) = 102,
J16(86) = 91, J(863)5 = - 8 867 J272(867) = 1, J(867)94 = - 2, J80(867) = 103, J26(867) = 19, J7(867) = 22 869 J26(869) = 1. J15(869) = 21, J9(869) = -12 871 J66(871) = 1, J29(871) = 4, J18(871) = 14, J(871)8 = 16
19
873 J96(873) = 1, J18(873) = - 8, J14(873) = - 44, J8(873) = - 404, J(873)3 = 55 877 J438(877) = 1, J219(877) = - 1, J29(877) = - 41, J(877)26 = - 3, J5(877) = 22 879 J146(879) = 1, J47(879) = 4, J24(879) = 115, J(879)8 = -14 881 j440(881) = 1, j220(881) = - 1, j92(881 = - 2, j56(881) = 8, J11(881) = - 18, J(881)1= - 88 883 J441(883) = 1, J218(883) = - 2, J(883)91 = - 3 J127(883) = 6, J11(883) = 111, J(883)5= 4 887 J886(887) = 1, J443(887) = -1, J178(887) = 3 , J52(887) = 2, J10(887) = 141 889 J42(889) = 1, J21(889) = - 1, J(889)3 = - 8 891 J18(891) = 1 893 J414(893) = 1, J130(893) = 101, J49(893) = 13, J21(893) = - 102, J16(893) = 118, J5(893) = - 16 897 J66(897) = 1, J16(897) = 211, J10(897) = 16, J(897)7 = 511 899 J420(899) = 1, J291(899) = 2, J130(899) = 5, J33(899) = 8, J16(899) = 103, J(899)7 = - 17 J(899)2 = 9 901 J208(901) = 1, J124(901) = 13, J83(901) = 201, J39(901) = 107, J10(901) = 121, J5(901) = - 11 903 J42(903) = 1. J29(903) = 103, J10(109) = 109, J(903)5 = 31 907 J151(907) = 1, J59(907) = 21, J21(907) = -11, J(907)8 = - 12 909 J4(909) =1 911 J455(911) = 1, J(911)40 = 2, J(911)11 = 12, J(911)1 = - 91 913 J82(913) = 1, J(913)12 = 12, J(913)9 = 131, J(913)3 = 21 917 J390(917) = 1, J195(917) = - 1, J67(917) = 31, J15(917) = 202, J7(917) = 115 919 J459(919) = 1, J132(919) = 2, J42(919) = 13, J(919)10 = 11, J3(919) = 81, J(919)1= 92 921 J153(921) = 1, J(921)69 = - 2, J41(921) = 112, J15(921) = 4, J11(921) = 7, J(721)5 = - 161 923 J210(923) = 1, J(923)26 = 3, J23(923) = 4, J(923)3 = 12 927 J34(927) = 1, J(927)5 = - 8 929 J464(929) = 1, J232(929) = -1, J(929)73 = - 2, J21(929) = 11, J13(929) = 8, J(929)4 = - 666 931 J126(931) = 1, J63(931) = - 1, J50(931) = 16, J25(931) = - 4, J(931)6 = - 202
20
933 J155(933) = 1, J80(933) = 13, J26(933) = 16, J13(933) = 4, J7(933) = 106 937 J936(937) = 1, J468(937) = -1, J152(937) = 2, J136(937) = - 3, J12(937) = 117, J(937)6 = 106 939 J312(639) = 1, J118(939) = - 2, J67(939) = 211, J24(939) = 103, J(939)8 = - 11 941 J940(941) = 1, J470(941) = - 1, J207(941) = - 2, J(941)109 = 3, J(941) 56= - 4, J7(941) = -7
J(941)1 = - 94 943 J110(943( = 1, J49(943) = - 4, J14(943) = - 701, J(943)6 = 119 947 J473(947) = 1, J(947)188 = - 2, J(947)133 = 3, J97(947) = 4, J22(947) = - 7, J(947)10 = - 11,
J(947)2 = 161 949 J24(949) = 1, J11(949) = - 22 951 J79(951) = 1, J40(951) = 31, J10(951) = 103, J(951)8 = 121 953 J952(953) = 1, J476(953) = - 1, J42(953) = - 2, J(953)11 = 3, J(953)5 = - 44 957 J28(957) = 1, 959 J24(959) = 1,
J11(951) = 901 J10(959) = - 311, J(959)7 = - 401
961 J465(961) = 1, J34(961) = 111, J(961)8 = - 11 963 J53(963) = 1, J10(963) = - 8, J(963)5 = 19 967 J332(967) = 1, J161(967) = -1, J47(967) = 3, J27(967) = - 81, J(967)10 = - 112, J(967)2 = - 29 969 J144(969) = 1, J98(960) = - 2, J(969)46 = - 2, J23(969) = 22, J6(969) = - 8 971 J970(971) = 1, J485( 971) = -1, J200(971) = - 2, J(971)146= - 11, J67(971) = 41, J23(971) = 6,
J9(971) = 114, J5(971) = - 13 973 J138(973) = 1, J69(973) = - 1, J13(973) = - 116, J(973)4 = 191 977 J976(977) = 1, J488(977) = - 1, J142(977) = -2, J(977)85 = 3, J40(977) = 31, J21(977) = - 13,
J14(977) = - 111, J(977)3 = 85 979 J44(979) = 1, J(979)6= 111 981 J108(981) = 1, J56(981) = 118, J9(981) = - 8
21
983 J982(983) = 1, J491(983) = - 1, J(983)117 = - 3, J90(983)= 41, J9(983) = - 2, J3(983) = 17 987 J138(987) = 1, J53(987) = - 2, J21(987) = - 8, J8(987) = 121 989 J462(989) = 1, J261(989) = - 2, J60(989) = 4, J28(989) = 6, J(989)4 = 9 991 J495(991) = 1, J(991)223 = 2, J49(991) = 4, J26(991) = - 6 J(991)4 = - 11 993 J110(993)= 1, J34(993) = 22, J(993)13 = 106, J(993)5 = 61 997 J166(997) = 1, J83(997) = -1, J35(997) = 4, J(997)13 = - 16, J3(997) = 3 999 J3(999) = 1
TABLE 3 (BASE - 8)
Here the eight digits are the same digits of the decimal system – 0,1,2,3,4,5,6 and 7. Each position in this
number represents a power of the base (8)
Divisor
d
Divisibility rule/test Jg(d) = m / J(d)g = m
2 J1(2) = 0 3 j2(3) = 1, 4 j1(4) = 0 5 j4(5) = 1,
j1(3) = - 1
j2(5) = - 1
6 j1(2) = 0 & j2(3) = 1, j1(3) = - 1 7 j1(7) = 1 10 j1(10) = 0 11 j2(11) = 1, j1(11) = - 1 12 j1(2) = 0 & j4(5) = 1, j2(5) = - 1 13 j12(13) = 1, j5(13) = - 1, j2(13) = - 2
22
14 j1(4) = 0 & j2(3) = 1 j1(3) = - 1 15 j4(15) = 1, 16 j1(2) = 0 & 17 j4(17) = 1, 20 j2(20) = 0
j2(15) = - 1 j1(7) = 1 j(17)1 = 2
21 j10(21) = 1, j4(21) = -1, j(21)2 = 4 22 J1(2) = 0 & 23 j6(23) = 1,
j2(11) = 1, j1(11) = - 1 j3(23) = - 1
24 J1(4) = 0 & J4(5) = 1, J2(5) = - 1 25 J2(25) = 1 26 J1(2) = 0 & 27 J13(27) = 1,
J12(13) = 1, J4(27) = 2,
J5(13) = - 1, J2(13) = -2 J(27)1 = 3
30 J1(10) = 0 & J2(3) = 1, j1(3) = - 1 31 J24(31) = 1, J12(31) = - 1, J7(31) = 2, J(31)1 = - 3 32 J1(2) = 0 & 33 J6(33) = 1, 34 J1(4) = 0 &
J4(15) = 1, J2(15) = -1 J3(33) = - 1 J1(7) = 1
35 J34(35) = 1, J16(35) = - 1, J5(35) = - 2, J(35)3 = - 3 36 J1(2) = 0 & 37 J5(37) = 1,
J4(17) = 1, j(17)1 = 2 J2(37) = 2
40 J2(40) = 0 41 J12(41) = 1, J5(41) = - 1, J(41)1 = - 4 42 J1(2) = 0 & J10(21) = 1, J4(21) = - 1, J(21)2 = 4 43 J4(43) = 1
23
44 J1(4) = 0 & J2(11) = 1, J1(11) = - 1 45 J14(45) = 1, J6(45) = - 1, J3(45) = - 6 46 J1(2) = o & J6(23) = 1, J3(23) = - 1 47 J4(47)= 1, J(47)1= 5 50 J1(10)= 0 & j4(5) = 1, j2(5) = -1 51 j24(51) = 1, j12(51) = -1, j7(51) = 2, j4(51) = - 4, j(51)3 = - 2 52 j1(2) = 0 & j2(25) = 1 53 j16(53) = 1, j7(53) = - 1, j5(53) = 2, j(53)2 = - 2 54 j1(4) = 0 & j12(13) = 1, j5(13) = - 1, j2(13) = - 2 55 j4(55) = 1 56 j1(2) = 0 & J13(27) = 1, J4(27) = 2, J(27)1 = 3 57 j27(57) = 1, j10(57) = 2, j3(57) = - 5 60 j2(20) = 0 & j2(3) = 1, j1(3) = -1 61 j7(61) = 1, j(61)1 = - 6 62 j1(2) = 0 & J24(31) = 1, J12(31) = - 1, J7(31) = 2, J(31)1 = - 3 63 j10(63) =1, 64 j1(4) = 0 &
j3(63) = 2, j4(15) = 1,
j(63)2 = 4 j2(15) = - 1
65 j64(65) = 1, j32(65) = -1, j11(65) = - 2, j6(65) = 6, j(65)3= - 3 66 j1(2) = 0 & J6(33) = 1, J3(33) = - 1 67 j24(67) = 1, j13(67) = - 3, j7(67) = 2, j(67)1 = 7 70 J1(10) = 0 & j1(7) = 1 71 j6(71) = 1, j3(71) = – 1, j(71)1 = - 7 72 j1(2) = 0 & J34(35) = 1, J16(35) = - 1, J5(35) = - 2, J(35)3 = - 3 73 j72(73) = 1, j35(73) = - 1, j12(73) = - 2, j7(73) = - 3
24
74 j1(4) = 0 & j4(17) = 1, j(17)1 = 2 75 j24(75) = 1, j12(75) = - 1, j4(75) = 11, j2(75) = 3 76 j1(2) = 0 & J5(37) = 1, J2(37) = 2 77 J2(77) = 1 100 J2(100)= 0 101 J4(101) = 1, J2(101) = - 1 102 j1(2) = 0 & J12(41) = 1, J5(41) = - 1, J(41)1 = - 4 103 J26(103) = 1, J13(103) = - 1, J5(103) = 5, J4(103) = 11, J2(103) = - 3 104 j1(4) = 0 & j10(21) = 1, j4(21) = -1, j(21)2 = 4 105 J26105) = 1, J17(105) = 2, J10(105) = 4, J(105)7 = 2 106 j1(2) = 0 & J4(43) = 1 107 J43(107) = 1, J14(107) = 2, J(107)6 = 6 110 J1(10)= 0 & J2(11) = 1, J1(11) = - 1 111 J3(111)= 1 112 j1(2) = 0 & J14(45) = 1, J6(45) = - 1, J3(45) = - 6 113 J24(113) = 1, J16(113) = 4, J7(113) =2 114 j1(4) = 0 & j6(23) = 1, j3(23) = - 1 115 J12(115) = 1, J(115)2 = - 6 116 j1(2) = 0 & J4(47)= 1, J(47)1= 5 117 J15(117) = 1, J7(117) = 22, J(117)6 = 22 120 j2(20) = 0 & j4(5) = 1, j2(5) = - 1 121 J22(121) =1, J11(121) = - 1, J(121)1 = - 12 122 j1(2) = 0 & j24(51) = 1, j12(51) = -1, j7(51) = 2, j4(51) = - 4, j(51)3 = - 2 123 J122(123) = 1, J51(123) = - 1, J16(123) = - 2, J7(123) = - 11, J(123)3 = 6
25
124 j1(4) = 0 & J2(25) = 1 125 J10(125) = 1, J3(125) = 2 126 j1(2) = 0 & j16(53) = 1, j7(53) = - 1, j5(53) = 2, j(53)2 = - 2 127 J34(127) = 1, J23(127) = 2, J12(127) = 4, J4(127) = 7 130 J1(10)= 0 & j12(13) = 1, j5(13) = - 1, j2(13) = - 2 131 J13(131) = 1, J10(131) = 4, J4(131) = 2 132 j1(2) = 0 & j4(55) = 1 133 J4(133) = 1 134 j1(4) = 0 & J13(27) = 1, J4(27) = 2, J(27)1 = 3 135 J12(135) = 1, J7(135) = 2, J4(135) = 4 136 j1(2) = 0 & j27(57) = 1, j10(57) = 2, j3(57) = - 5 137 J14(137) = 1, J5(137) = - 7, J(137)1 = 14 140 J2(40) = 0 & j2(3) = 1, j1(3) = - 1 141 J20(141) = 1, J10(141) = -1, J(141)3 = 22 142 j1(2) = 0 & j7(61) = 1, j(61)1 = - 6 143 J12(143)= 1, J5(143) = - 1, J2(11) = 1 & J2(13) = - 2 144 j1(4) = 0 & J24(31) = 1, J12(31) = - 1, J7(31) = 2, J(31)1 = - 3
TABLE 4 ( BASE- 9 ) Here the nine digits are the same digits of the decimal system – 0,1,2,3,4,5,6,7and 8. Each position in this number
system represents a power of the base (9)
Divisor d
Divisibility rule/test Jg(d) = m / J(d)g = m
2 J1(2) = 1 3 J1(3) = 0
26
4 J1(4) = 1 5 J2(5) = 1 6 J1(3) = 0 & 7 J3(7) = 1 8 J1(8) = 1 10 J1(10) = 0 11 J2(11) = 1 12 J5(12) = 1, 13 J1(3) = 0 & 14 J3(14) = 1, 15 J1(2) = 1 & 16 J1(3) = 0 &
J1(2) = 1 J3(12) = 3, J(12)2 = 3
J1(4) = 1
J(14)1 = 3
J3(7) = 1
J2(5) = 1
17 J2(17) = 1 18 J8(18) = 1, 20 J1(10) = 0 & 21 J10(21) = 1, 22 J1(4) = 1 & 23 J1(3) = 0 & 24 J1(2) = 1 & 25 J12(25) = 1,
J4(18) = - 1, J1(2) = 1
J(21)2 = 4 J2(5) = 1
J3(7) = 1
J5(12) = 1,
J6(25) = 3,
J(18)1 = 2 J3(12) = 3, J(12)2 = 3 J(25)2 = 2
26 J1(3) = 0 & 27 J11(27) = 1, 28 J1(2) = 1 &
J1(8) = 1 J5(27) = - 1,
J3(14) = 1,
J2(27) = - 4 J(14)1 = 3
30 J2(30) = 0
27
31 J1(4) = 1 & 32 J15(32) = 1,
J3(7) = 1 J7(32) = - 1, J3(32) = 4
33 J1(3) = 0 & J2(11) = 1 34 J16(34) = 1, J9(34) = 4, J(34)3 = 2 35 J4(35) = 1 36 J1(3) = 0 & J5(12) = 1, J3(12) = 3, J(12)2 = 3 37 J1(2) = 1 & 38 J2(5) = 1 & 40 J1(10) = 0 & 41 J10(41) = 1,
J8(18) = 1, J3(7) = 1 J1(4) = 1 J(41)4 = - 3,
J4(18) = - 1, J(18)1 = 2
J2(41) = 7
42 J1(2) = 1 & 43 J1(3) = 0 & 44 J2(5) = 1 & 45 J4(45) = 1, 46 J1(3) = 0 &
J10(21) = 1, J3(14) = 1, J1(8) = 1
J2(45) = - 1
J1(2) = 1 &
J(21)2 = 4 J(14)1 = 3
J3(7) = 1
47 J23(47) = 1, J12(47) = - 3, J6(47) = 4, J3(47) = - 2 48 J1(4) = 1 & J5(12) = 1, J3(12) = 3, J(12)2 = 3 50 J1(10) = 0 & J2(5) = 1 51 J1(2) = 1 & 52 J25(52) = 1,
J12(25) = 1, J13(52) = 3,
J6(25) = 3, J(25)2 = 2 J(52)3 = 2
53 J1(3) = 0 & J2(17) = 1 54 J23(54) = 1, 55 J1(2) = 1 & 56 J1(3) = 0 &
J14(54) = 2, J11(27) = 1, J8(18) = 1,
J5(54) = 4, J5(27) = - 1, J4(18) = - 1,
J3(54) = - 6, J(54)1 = 12 J2(27) = - 4 J(18)1 = 2
28
57 J1(4) = 1 & J3(14) = 1, J(14)1 = 3 58 J28(58) = 1, J14(58) = - 1, J6(58) = 11, J(58)3 = 4 60 J2(30) = 0 & J1(2) = 1 61 J2(5) = 1& J5(12) = 1, J3(12) = 3, J(12)2 = 3 62 J3(7) = 1 & 63 J1(3) = 0 & 64 J1(2) = 1 & 65 J32(65) = 1, 66 J1(3) = 0 &
J1(8) = 1 J10(21) = 1, J15(32) = 1, J16(65) = 3,
J1(4) = 1 &
J(21)2 = 4 J7(32) = - 1,
J9(65) = - 2,
J2(5) = 1
J3(32) = 4 J(65)4 = 5
67 J5(67) = 1, J3(67) = -3 68 J1(2) = 1 & 70 J1(10) = 0 & 71 J8(71) = 1, 72 J2(5) = 1&
J16(34) = 1, J3(7) = 1
J(71)1 = - 7
J3(14) = 1,
J9(34) = 4, J(34)3 = 2
J(14)1 = 3
73 J1(3) = 0 & J1(2) = 1 & J5(12) = 1, J3(12) = 3, J(12)2 = 3 74 J12(74) = 1, 75 J1(4) = 1 & 76 J1(3) = 0 &
J6(74) = - 3, J8(18) = 1, J12(25) = 1,
J4(74) = - 5 J4(18) = - 1, J6(25) = 3,
J(18)1 = 2 J(25)2 = 2
77 J3(7) = 1 & J2(11) = 1 78 J38(78) = 1, J25(78) = 2, J20(78) = 3, J(78)13 = 2, J12(78) = 4, J6(78) = 6, 80 J1(10) = 0 & 8 1 J6(81) = 1, 82 J1(2) = 1 & 83 J1(3) = 0 &
J1(8) = 1 J3(81) = - 1,
J10(41) = 1, J11(27) = 1,
J(8)1 = - 8 J(41)4 = - 3, J5(27) = - 1,
J2(41) = 7 J2(27) = - 4
29
84 J1(4) = 1 & 85 J3(7) = 1 & 86 J1(3) = 0 & 87 J43(87) = 1,
J10(21) = 1, J5(12) = 1,
J1(2) = 1 &
J22(87) = -3,
J(21)2 = 4 J3(12) = 3,
J3(14) = 1, J(87)8 = 5,
J(12)2 = 3 J(14)1 = 3
J4(87) = 4, J2(87) = 2
88 J2(5) = 1 & 100 J2(100) = 0 101 J1(2) = 1 &
J2(17) = 1
J4(45) = 1, J2(45) = - 1
102 J45(102) = 1, J23(102) = 3, J14(102) = - 5, J4(102) = 4, J2(102) = - 2 103 J1(3) = 0 & 104 J2(5) = 1 & 105 J1(2) = 1 & 106 J1(3) = 0 &
J1(4) = 1 & J8(18) = 1,
J23(47) = 1, J15(32) = 1,
J3(7) = 1 J4(18) = - 1,
J12(47) = - 3,
J7(32) = - 1,
J(18)1 = 2 J6(47) = 4, J3(47) = - 2
J3(32) = 4
107 J1(8) = 1 & J5(12) = 1, J3(12) = 3, J(12)2 = 3 108 J48(108) = 1, J24(108) = - 1, J8(108) = 2, J(108)1 = 11 110 J1(10) = 0 & 111 J3(7) = 1 & 112 J1(4) = 1 & 113 j1(3) = 0 & 114 J1(2) = 1 & 115 J2(5) = 1&
J2(11) = 1 J3(14) = 1, J12(25) = 1, J16(34) = 1,
J25(52) = 1, J10(21) = 1,
J(14)1 = 3 J6(25) = 3,
J9(34) = 4,
J13(52) = 3,
J(21)2 = 4
J(25)2 = 2 J(34)3 = 2 J(52)3 = 2
116 j1(3) = 0 & J4(35) = 1 117 J26(117) = 1, J13(117) = - 1, J7(117) = - 4, J(117)2 = 6 118 J1(2) = 1 & J23(54) = 1, J14(54) = 2, J5(54) = 4, J3(54) = - 6, J(54)1 = 12 120 J1(10) = 0 & J5(12) = 1, J3(12) = 3, J(12)2 = 3
30
121 J1(4) = 1 & J11(27) = 1, J5(27) = - 1, J2(27) = - 4
TABLE 5 ( BASE- 11) Here the first 10 digits are the same digits of the decimal system – 0,1,2,3,4,5,6,7,8 and 9.The remaining single
digit is the symbol A representing decimal value 10. Each position in this number system represents a power of
the base (11)
Divisor d
Divisibility rule/test Jg(d) = m / J(d)g = m
2 J1(2) = 1 3 J2(3) = 1 4 J2(4) = 1, 5 J1(5) = 1 6 J2(6) = 1, 7 J3(7) = 1, 8 J2(8)= 1 9 J6(9) = 1, A J1(A) = 1 10 J1(10) = 0
J1(3) = - 1 J1(4) = - 1
J1(6) = - 1 J2(7) = 2
J3(9) = - 1
11 J2(11) = 1 12 J11(12) = 1, 13 J3(13) = 1 14 J2(14) = 1 15 J4(15) = 1,
J6(12) = - 1, J(12)2 = - 3
J(15)1= 3
16 J15(16) = 1, 17 J6(17) = 1,
J8(16) = - 1, J3(17) = - 1,
J2(16) = 2, J(16)1 = - 3 J(17)1 = 5
31
18 J3(18) = 1 19 J2(19) = 1 1A J6(1A) = 1, 20 J1(10) = 0 &
J(1A)1= 2 J1(2) = 1
21 J20(21) = 1, J10(21) = - 1, J3(21) = - 3, J(21)1 = - 2 22 J2(22) = 1 23 J5(23) = 1, 24 J11(24) = 1, 25 J17(25) = 1,
J3(23) = 6, J6(24) = - 1, J9(25) = - 1,
J(23)2 = 6 J4(24) = 3, J7(25) = 2,
J(24)2 = - 3 J(25)2 = - 2
26 J6(26) = 1, J(26)2 = - 3 27 J26(27) = 1, J13(27) = - 1, J9(27) = 2, J(27)5 = - 2 J3(27) = - 3 28 J2(28) = 1 29 J28(29) = 1, 2A J8(2A) = 1,
J14(29) = - 1, J3(29) = - 2 J(2A)1 = 3
30 J1(10) = 0 & 31 J15(31) = 1 ,
J2(3) = 1, J8(31) = - 1,
J1(3) = - 1 J3(31) = 5, J(31)1 = - 3
32 J3(32) = 1 33 J6(33) = 1, 34 J6(34) = 1, 35 J3(35) = 1 36 J11(36) = 1 ,
J3(33) = - 1 J3(34) = - 1
J7(36) = 2, J(36)2 = A
37 J2(37) = 1 38 J37(38) = 1 , 39 J6(39) = 1,
J19(38) = - 1, J1(6) = - 1 &
J5(38) = 3, J2(38)= - 2 J2(7) = 2
32
3A J7(3A) = 1, J3(3A) = - 2 40 J1(10) = 0 & J2(4) = 1, J1(4) = - 1 41 J6(41) = 1, J(41)1 = - 4, J1(5) = 1 & J3(9) = - 1 42 J20(42) = 1, J10(42) = - 1, J3(42) = - 3 43 j42(43) = 1, j21(43) = - 1, j18(43) = -2, j(43)7 = -3, j(43)4 = 2 44 j4(44) = 1 45 J1A(45) = 1, 46 j5(46) = 1,
j8(45) =2, j1(2) = 1 &
j5(45) = -11 j(23)2 = 6
47 j15(47) = 1, 48 j11(48) = 1,
jA(47) = -2, j5(48) = 7,
j4(47) = 4 j(48)2 = -3
49 j24(49) = 1, 4A j17(4A) = 1,
j12(49) = - 1, J9(4A) = - 1,
j9(49) = 4, j4(4A) = 7,
j(49)4 = - 4 J(4A)1 = 5
50 J1(10) = 0 & J1(5) = 1 51 J6(51) = 1, 52 J6(52) = 1, 53 j26(53) = 1,
J(51)1 = - 5 J2(52) = 7 j13(53) = - 1, j3(53) = - 3
54 j53(54) = 1, j27(54) = - 1, j7(54) = 2, J2(54) = 3 55 J2(55) = 1 56 J4(56) = 1, J2(56) = - 1 57 J28(57) = 1, J14(57) = - 1, J(57)5 = - 5 J2(57) = - 3 58 J6(58) = 1, 59 J15(59) = 1,
J2(58) = - 5 J7(59) = 3, J(59)3 = - 5
5A j11(5A) = 1, j8(5A) = - 4, j(5A)4 = - 4 60 J1(10) = 0 & J2(6) = 1, J1(6) = - 1,
33
61 j60(61) = 1, j30(61) = - 1, j13(61) = - 2, j12(61) = 11, j(61)5 = - 4 62 j15(62) = 1, 63 j20(63) = 1,
j7(62) = 3, j10(63) = - 1,
j(62)3 = 7 j5(63) = 5, j(63)2 = 4
64 j3(64) = 1 65 j64(65) = 1, j32(65) = - 1, j15(65) = 2, j8(65) = 11, j(65)3 = - 4 66 j6(66) = 1, j2(8) = 1 & j3(9) = - 1 67 j66(67) = 1, 68 j6(68) = 1, 69 jA(69) = 1, 6A j6(6A) = 1, 70 j1 (10) = 0 & 71 j11 (71) = 1, 72 j36(72) = 1,
j33(67) = - 1, j3(68) = - 1, j(69)3 = - 4 j(6A)1 = 7
j3( 7 ) = 1,
j3 ( 71) = 5
j21(72) = 2,
j8(67) = 2,
j2 (7) = 2
j7(72) = 4,
j6(67)= - 3
j3(72) = - 11
73 j4(73) = 1, J1(5) = 1 & J(15)1= 3 74 j4A(74) = 1, 75 j37(75) = 1, 76 j38(76) = 1,
j25(74) = - 1, j19(75) = - 1, j25(76) = 11,
j8(74) = - 5, j5(75) = 3 j11(76) = - 2,
j(74)2 = - 2
j3(76) = 3
77 j6(77) = 1, J2(11) = 1 & J3(7) = 1, J2(7) = 2 78 j15(78) = 1, j(78)4 = - 4 79 j7(79) = 1 OR j1(2) = 1 & j7(3A) = - 1, j3(3A) = - 2 7A j26(7A) = 1, 80 j1 (10) = 0 &
j9(7A) = 2, j(7A) 4 = 7 j2( 8 ) = 1
81 j20 (81) = 1, j10( 81) = - 1, j7 (81) = - 2, j3(81) = - 4 82 j6(82) = 1, J1(A) = 1 & J3(9) = - 1
34
83 j11(83) = 1, j(83)2 = - 3 84 j20(84) = 1, 85 J28(85) = 1, 86 J42(86) = 1,
j10( 84) = - 1, J14(85) = - 1, J21(86) = - 1,
j6(84) = 9, J6(85) = 4,
JA(86) = - 5,
j( 84)4 = - 7 J(85) 2 = A J(86)7 = - 3
87 J3(87) = 1 88 J8(88) = 1, J2(8)= 1 & J2(11) = 1 89 j44(89) = 1, 8A j1A(8A) = 1,
j22(89) = - 1, jA(8A) = - 3,
j10(89) = - 2, j(89)2 = - 4 j(8A)4 = - 5
90 j1(10) = 0 & 91 jA(91) = 1,
j6(9) = 1, j3(9) = - 1 j(91)! = - 1
TABLE 6 (BASE -12)
Here the first 10 digits are the same digits of the decimal system – 0,1,2,3,4,5,6,7,8 and 9.The remaining two
digit are the symbol A and B representing decimal values 10 and 11. Each position in this number system
represents a power of the base (12).
Divisor
d
Divisibility rule/test Jg(d) = m / J(d)g = m
2 j1(2) = 0 3 j1(3) = 0 4 j1(4) = 0 5 j4(5) = 1, 6 j1(6) = 0 7 j6(7) = 1, 8 j2(8) = 0
j2(5) = - 1
j3(7) = - 1
35
9 j2(9) = 0 A j1(2) = 0 & J4(5) = 1, j2(5) = - 1 B j1(B) = 1 10 j1(10) = 0 11 j2(11) = 1 12 j1(2) = 0 & 13 j1(3) = 0 &
j6(7) = 1, j4(5) = 1,
j3(7) = - 1 j2(5) = - 1
14 j2(14) = 0 15 j14(15) = 1, j8(15) = - 1, j6(15) = 2, j(15)2 = - 2 16 j1(2) = 0 & 17 j6(17) = 1, 18 j1(4) = 0 & 19 j1(3) = 0 & 1A j1(2) = 0 & 1B jB(1B) = 1,
j2(9) = 0 j3(17) = - 1 j4(5) = 1, j6(7) = 1, j1(B) = 1
j3(1B) = 3,
j2(5) = - 1 j3(7) = - 1
j(1B)1 = 2
20 j2(20) = 0 21 j18(21) = 1, jA(21) = - 1, j3(21) = 3, j(21)1 = - 2 22 j1(2) = 0 & 23 j3(23) = 0 24 j1(4) = 0 &
j2(11) = 1
j6(7) = 1 j3(7) = - 1
25 j4(25) = 1, j2(25) = - 1 26 j1(6) = 0 & j4(5) = 1 j2(5) = - 1 27 j26(27) = 1, j13(27) = - 1, j6(27) = 2, j(27)3 = - 4 28 j3(28) = 0
36
29 j1(3) = 0 & j1(B) = 1 2A j1(2) = 0 & j14(15) = 1, j8(15) = - 1, j6(15) = 1, j(15)2 = - 2 2B j10(2B) = 1, j7(2B) = - 2, j(2B)1 = 3 30 j2(30) = 0 31 j9(31) = 1, 32 j1(2) = 0 & 33 j1(3) = 0 &, 34 j2(8) = 0 &
j(31)3 = A, j6(17) = 1,
j2(11) = 1 j4(5) = 1,
j(31)1 = - 3 j3(17) = - 1
j2(5) = - 1
35 j34(35) = 1, j18(35) = - 1, j5(35) = 3 j(35) 2 = 2 36 j1(6) = 0, j6(7) = 1, j3(7) = - 1 37 j36(37) = 1, j19(37) = - 1, j13(37) = 2, j(37)6 = - 2, j(37)1 = - 21 38 j1(4) = 0 & 39 j2(9) = 0 & 3A j1(2) = 0 &
j1(B) = 1 j4(5) = 1, jB(1B) = 1,
j2(5) = - 1
j3(1B) = 3, j(1B)1 = 2
3B j1B(3B) = 1, jB(3B) = 2, j7(3B) = - 5, j2(3B) = 3 40 j1(3) = 0 & j2(14) = 0 41 j36(41) = 1, 42 j1(2) = 0 & 43 j1(3) = 0 &
j19(41) = - 1, j18(21) = 1, j14(15) = 1,
jA(41) = 2, jA(21) = - 1, j8(15) = - 1,
j5(41) = A j3(21) = 3, j6(15) = 2,
j2(41) = - 3 j(21)1 = - 2
j(15)2 = - 2
44 j1(4) = 0 & j1(B) = 1 45 j44(45) = 1, j22(45) = - 1, jB(45) = 2, j5(45) = - 3 46 j3(46) = 0 47 j4(47) = 1 48 j2(8) = 0 & j6(7) = 1, j3(7) = - 1
37
49 j1(3) = 0 & 4A j1(2) = 0 & 4B j25(4B) = 1, 50 j1(10) = 0 & 51 j13(51) = 1, 52 j1(2) = 0 &
j6(17) = 1, j4(25) = 1,
jB(4B) = 3,
j4(5) = 1,
j4 (51) = - 4, j26(27) = 1,
j3(17) = - 1 j2(25) = - 1
j(4B)5 = - 2,
j2(5) = - 1
j(51)3 = - 3 j13(27) = - 1,
j(4B)1 = 5
j6(27) = 2, j(27)3 = - 4
53 j2(9) = 0 & 54 j3(54) = 0 55 j4(55) = 1 56 j1(6) = 0 &
j6(7) = 1, j3(7) = - 1
j1(B) = 1
57 J56(57) = 1, j29(57) = - 1, j9(57) = 3, j(57)4 = - 2 58 j1(4) = 0 & 59 j1(3) = 0 & 5A j1(2) = 0 &
j14(15) = 1, jB(1B) = 1, j10(2B) = 1,
j8(15) = - 1, j3(1B) = 3, j7(2B) = - 2,
j6(15) = 2, j(15)2 = - 2 j(1B)1 = 2
j(2B)1 = 3
5B J2B(5B) = 1, j1B(5B) = A, j(5B)3 = 3 60 j2(60) = 0 61 J30(61) = 1, j16(61) = - 1, j(61)3 = 3 62 j1(2) = 0 & 63 j1(3) = 0 &
j9(31) = 1, j18(21) = 1,
j(31)3 = A, jA(21) = - 1,
j(31)1 = - 3 j3(21) = 3, j(21)1 = - 2
64 j1(4) = 0 & 65 J1 (B) = 1 & 66 j1(6) = 0 &
j6(17) = 1,
j6(7) = 1 j2(11) = 1
j3(17) = - 1 j3(7) = - 1
67 j22(67) = 1, J11(67) = - 1, j(67)3 = - 8 68 j2(14) = 0 & j4(5) = 1, j2(5) = - 1
38
69 j4(69) = 0 6A j1(2) = 0 & 6B j35(6B) = 1,
j34(35) = 1,
JA(6B) = 4,
j18(35) = - 1,
J5(6B) = - 2,
j5(35) = 3 j(6B)9 = 3
j(35) 2 = 2
j(6B)3 = B
70 j1(10) = 0 & 71 j14(71) = 1,
j6(7) = 1, J4(71) = - 4,
j3(7) = - 1
j(71)3 = - 3
72 j1(2) = 0 & j36(37) = 1, j19(37) = - 1, j13(37) = 2, j(37)6 = - 2, j(37)1 = - 21 73 j1(3) = 0 & 74 j2(8) = 0 & 75 j8(75) = 1, 76 j1(2) = 0 &
j4(25) = 1, J1 (B) = 1
J4(75) = - 1
j2(9) = 0 &
j2(25) = - 1
J4(5) = 1, j2(5) = - 1
77 j6(77) = 1, J3(77) = - 1 78 j1(4) = 0 & jB(1B) = 1, j3(1B) = 3, j(1B)1 = 2 79 j1(3) = 0 & 7A j1(2) = 0 &
j26(27) = 1, j1B(3B) = 1,
j13(27) = - 1, jB(3B) = 2,
j6(27) = 2, j7(3B) = - 5,
j(27)3 = - 4 j2(3B) = 3
7B jC(7B) = 1, J8(7B) = B, j(7B)4 = B OR j4(5) = 1, j2(5) = - 1 & j6(17) = 1, j3(17) = - 1 80 j1(3) = 0 & 81 j14(81) = 1,
j3(28) = 0 J8(81) = - 1, J(81)1 = - 8
82 j1(2) = 0 & j36(41) = 1, j19(41) = - 1, jA(41) = 2, j5(41) = A j2(41) = - 3 83 j2(9) = 0 & J1 (B) = 1 84 j1(4) = 0 & j18(21) = 1, jA(21) = - 1, j3(21) = 3, j(21)1 = - 2
TABLE 7 (BASE - 13)
39
Here the first 10 digits are the same digits of the decimal system – 0,1,2,3,4,5,6,7,8 and 9.The remaining three
digits are the symbol A,B and C representing decimal values 1o,11and 12. Each position in this number system
represents a power of the base (13)
Divisor d
Divisibility rule/test Jg(d) = m / J(d)g = m
2 j1(2) = 1 3 j1(3) = 1 4 j1(4) = 1 5 j4(5) = 1, 6 j1(6) = 1 7 j2(7) = 1, 8 j2(8) = 1 9 j3(9) = 1 A j4(A) = 1 B jA(B) = 1,
j2(5) = - 1
j1(7) = - 1
j5(B) = - 1, j2(B) = 4
C j1(C) = 1 10 j1(10) = 0 11 j2(11) = 1 12 j4(12) = 1, 13 j4(13) = 1 14 j4(14) = 1, 15 j3(15) = 1
j2(12) = 4 j(13)1 = 5 j2(14) = - 1
16 j15(16) = 1, 17 j4(17) = 1, 18 j2(18) = 1 19 jA(19) = 1,
j9(16) = - 1, j(17)1 = - 3
j5(19) = - 1,
j4(16) = 4, j(16)2 = - A
j(19)2 = 3
40
1A JB(1A) = 1, j5(1A) = 4, j(1A)3 = 2 1B j2(1B) = 1 1C j17(1C) = 1, jA(1C) = - 1, j3(1C) = - 3, j(1C)1 = 2 20 j1(10) = 0 & 21 j9(21) = 1,
j1(2) = 1 j4(21) = - 5, j(21)1 = - 2
22 j2(22) = 1 23 j11(23) = 1, j7(23) = - 1, j(23)3 = 4 24 j4(24) = 1 25 j24(25) = 1, 26 j8(26) = 1, 27 jA(27) = 1,
j12(25) = - 1, j5(26) = - 3, j6(27) = - 2,
j9(25) = - 2 j(25)4= - 3 j(26)3 = - 3 j2(27) = 4
28 j4(28) = 1, 29 j4(29) = 1, 2A j3(2A) = 1
j2(28) = - 1 j(29)1 = - 8
2B j2A(2B) = 1, 2C j15(2C) = 1,
j15(2B) = - 1, j9(2C) = - 1,
j5(2B) = - 2, j5(2C) = - 5,
J(2B)2 = -7 j(2C)1 = 3
30 j1(10) = 0 & 31 j4(31) = 1,
j1(3) = 1 j(31)1 = - 3
32 j31(32) = 1, j17(32) = - 1, j6(32) = 2, j(32)2 = - 8 33 j2(33) = 1 34 j18(34) = 1, 35 jA(35) = 1, 36 jC(36) = 1, 37 jB(37) = 1,
jc(34) = - 2, j3(35) = - 3 j5(36) = - 2, j4(37) = - 5,
j3(34) = 4
j(36) 2 = 4 j(37) 2 = 3
41
38 j37(38) = 1, j1A(38) = - 1, jA(38) = 2, j6(38) = 3, j(38)3 = - 4 39 j4(39) = 1, 3A j11(3A) = 1, 3B j17(3B) = 1, 3C j4(3C) = 1, 40 j1(10) = 0 & 41 j10(41) = 1, 42 j9(42) = 1,
j(39)1 = - B
j7(3A) = - 1, jA(3B) = - 1,
j(3C)1 = 4
j1(4) = 1 j4(41) = - 6,
j4(42) = - 5,
j(3A)3 = 6
j3(3B) = - 3
j(41)1 = - 4 j2(42) = 7
43 j17(43) = 1, jB(43) = 2, j3(43) = - 3 44 j2(44) = 1 45 j15(45) = 1, 46 j11(46) = 1,
j4(45) = 4, j7(46) = - 1,
j(45)4 = - 11, j2(45) = - 2
j2(46) = - 5 47 j46(47) = 1, j23(47) = - 1, j(47)9 = 2, j4(47) = 5 48 j1(3) = 1 & j1(4) = 1 & j4(5) = 1 j2(5) = - 1 49 j3(49) = 1 4A j24(4A) = 1, j12(4A) = - 1, j8(4A) = 7, j(4A)4 = - 3 4B j6(4B) = 1 then j3(9) = 1 & j2(7) = 1 4C j13(4C) = 1, j6(4C) = - 7, j(4C)3 = - 3 50 j1(10) = 0 & j4(5) = 1, j2(5) = - 1 51 j1(6) = 1 & jA(B) = 1, j5(B) = - 1, j2(B) = 4 OR jA(51) = 1, j(51)3 = 7, j(51)1 = - 5 52 j51(52) = 1, j27(52) = - 1, j9(52) = 3 j7(52) = 2, j(52)4 = - 7 53 j4(53) = 1, 54 jB(54) = 1,
j(53)1 = 21 OR j5(54) = 4 OR
j4(14) = 1, j(1A)3 = 2 &
j2(14) = - 1 & j1(4) = 1 j1(3) = 1
55 j4(55) = 1 OR j2(11) = 1 & j4(5) = 1, j2(5) = - 1
42
56 j55(56) = 1, j29(56) = - 1, j16(56) = - 2, j7(56) = - 5, j3(56) = - 4 57 j6(57) = 1 OR j2(8) = 1 & j3(9) = 1 58 j57(58) = 1, j2A(58) = - 1, j13(58) = 2, jB(58) = 5, j(58)6 = 3, j3(58) = 7 59 j1(2) = 1 & j2A(2B) = 1, j15(2B) = - 1, j5(2B) = - 2, J(2B)2 = -7 5A j1(3) = 1 & j17(1C) = 1, jA(1C) = - 1, j3(1C) = - 3, j(1C)1 = 2 OR j17(5A) = 1, j9(5A) = - 2, j(5A)2=4 5B j1(4) = 1 & j15(16) = 1, j9(16) = - 1, j4(16) = 4, j(16)2 = - A OR
J15(5B) = 1, j8(5B) = - 3, j3(5B) = - 7 5C JA(5C) = 1, j5(5C) = - 1, j(5C)1 = 6 OR j2(7) = 1, j1(7) = - 1 & jA(B) = 1, j5(B) = - 1, j2(B) = 4 60 j1(10) = 0 & j1(6) = 1 61 j30(61) = 1, j1A(61) = 4, j7(61) = 2, j(61)1 = - 6 62 j4(62) = 1 63 j21(63) = 1, 64 j31(64) = 1, 65 j64(65) = 1,
j14(63) = - 2, j17(64) = - 1, j32(65) = -1,
j7(63) = 4, j5(64) = - 3, j8(65) = - 2,
j(63)3 = - 8
j2(64) = 5 j2(65) = 3
66 j2(66) = 1 67 j4(67) = 1, j2(67) = - 1 68 j1(2) = 1 & 69 j1(3) = 1 &
j18(34) = 1, j11(23) = 1,
jc(34) = - 2, j7(23) = - 1,
j3(34) = 4 OR j(23)3 = 4 OR
j18(68) = 1, j11(69) = 1,
j8(68) = -5, j(69)3 = 4,
j2(68) = - 3 j2(69) = - 5
6A jA(6A) = 1, j3(6A) = - 3 OR j2(8) = 1 & jA(B) = 1, j5(B) = - 1, j2(B) = 4 6B j6A(6B) = 1, j35(6B) = -1, j13(6B) = 2, j(6B)5 = - 6 6C j3(9) = 1 & 70 j1(10) = 0 &
j4(A) = 1
j2(7) = 1, j1(7) = - 1
71 j1(4) = 1 & JB(1A) = 1, j5(1A) = 4, j(1A)3 = 2 72 j1(3) = 1 & j24(25) = 1, j12(25) = - 1, j9(25) = - 2, j(25)4= - 3 OR j24(72) = 1, j9(6B) = -2, j(72)5 = - 5
43
73 j1(2) = 1 & 74 j4(5) = 1,
j37(38) = 1, j2(5) = - 1 &
j1A(38) = - 1, j15(16) = 1,
jA(38) = 2, j9(16) = - 1,
j6(38) = 3, j4(16) = 4,
j(38)3 = - 4 j(16)2 = - A
75 j1(3) = 1 & j8(26) = 1, j5(26) = - 3, j(26)3 = - 3 OR j8(75) = 1, j3(75) = - B 76 j75(76) = 1, j39(76) = -1, j28(76) = 2, j19(76) = 3, j(76)11 = 2, j(76)7 = - 11, j(76)1 = 12 77 j1(2) = 1 & j11(3A) = 1, j7(3A) = - 1, j(3A)3 = 6 78 j3(9) = 1 & jA(B) = 1, j5(B) = - 1, j2(B) = 4 79 j1(4) = 1 & j17(1C) = 1, jA(1C) = - 1, j3(1C) = - 3, j(1C)1 = 2
TABLE 8 (BASE- 14)
Here the first 10 digits are the same digits of the decimal system – 0,1,2,3,4,5,6,7,8 and 9.The remaining four
digits are the symbol A,B ,C and D representing decimal values 10,11, 12and 13. Each position in this number
system represents a power of the base (14)
Divisor d
Divisibility rule/test Jg(d) = m / J(d)g = m
2 j1(2) = 0 3 j2(3) = 1, 4 j2(4) = 0, 5 j2(5) = 1, 6 j1(2) = 0 &
j1(3) = - 1 j1(4) = 2 j1(5) = - 1
j2(3) = 1, j1(3) = - 1
7 j1(7) = 0 8 j3(8) = 0, 9 j6(9) = 1
j2(8) = 4 j3(9) = -1
A j1(2) = 0 & j2(5) = 1, j1(5) = - 1
44
B j5(B) = 1, j2(B) = - 2 C j2(4) = 0, j1(4) = 2 & j2(3) = 1, j1(3) = - 1 D j1(D) = 1 10 j1(10) = 0 11 j2(11) = 1, 12 j4(12) = 0, 13 j12(13) = 1,
j1(11) = - 1 j2(12) = 4 j8(13) = - 1, j(13)2 = 2
14 j1(2) = 0 & j6(9) = 1 j3(9) = -1 15 j14(15) = 1 j9(15) = - 1, j4(15) = - 2, j(15)1 = - 4 16 j2(4) = 0, 17 j1(7) = 0 & 18 j1(2) = 0 &
j1(4) = 2 &
j2(3) = 1, j5(B) = 1,
j2(5) = 1, j1(5) = - 1 j1(3) = - 1
j2(B) = - 2
19 j18(19) = 1, jB(19) = -1, j6(19) = 3, j(19)2 = 2 1A j3(8) = 0, j2(8) = 4 & j2(3) = 1, j1(3) = - 1 1B jA(1B) = 1, j5(1B) = - 1, j2(1B) = - 4 1C j1(2) = 0 & j12(D) = 1, j8(D) = - 1, j(D)2 = 2 1D j14(1D) = 1 j9(1D) = - 1, j4(1D) = - 5, j(1D)1 = 2 20 j3(20) = 0 21 j20(21) = 1, j10(21) = - 1, j9(21) = 3, j6(21) = 5, j(21)1 = - 2 22 j2(3) = 1, j1(3) = - 1 & j1(2) = 0 & j2(5) = 1, j1(5) = - 1 23 j11(23) = 1, j9(23) = 4, j5(23) = 5, j(23)3 = 2 24 j5(24) = 0, j2(24) = 4 25 j2(3) = 1, j1(3) = - 1 & j5(B) = 1, j2(B) = - 2 26 j1(2) = 0 & j12(13) = 1, j8(13) = - 1, j(13)2 = 2
45
27 j1(7) = 0 & 28 j2(4) = 0 &
j2(5) = 1, j6(9) = 1
j1(5) = - 1 j3(9) = -1
29 jC(29) = 1 j6(29) = - 1, j3(29) = 6 2A j1(2) = 0 & j14(15) = 1 j9(15) = - 1, j4(15) = - 2, j(15)1 = - 4 2B j1(D) = 1 & j2(3) = 1 2C j2(5) = 1, j1(5) = - 1 & j3(8) = 0, j2(8) = 4 2D j8(2D) = 1, 30 j1(2) = 0 &
j4(2D) = - 1, j1(7) = 0 &
j(2D)1 = 3 j2(3) = 1
31 j17(31) = 1 j11(31) = - 2, j9(31) = 4, j(31)4 = - 5, j(31)1 = - 3 32 j2(4) = 0 & 33 j6(33) = 1
j5(B) = 1, j2(B) = - 2 j3(33) = -1
34 j1(2) = 0 & 35 j19(35) = 1,
j18(19) = 1, j9(35) = 4,
jB(19) = -1, j5(35) = 3,
j6(19) = 3, j(35)7 = 2,
j(19)2 = 2 j2(35) = 8
36 j2(3) = 1 & j4(12) = 0, j2(12) = 4 37 j2(37) = 0 38 j1(2) = 0 & 39 j2(3) = 1 &
jA(1B) = 1, j12(13) = 1,
j5(1B) = - 1, j8(13) = - 1,
j2(1B) = - 4 j(13)2 = 2
3A j2(4) = 0, j1(4) = 2 & j1(D) = 1 3B j3A(3B) = 1, j1C(3B) = - 1, j7(3B) = 2, j(3B)4 = - 6 3C j1(2) = 0 & j14(1D) = 1 j9(1D) = - 1, j4(1D) = - 5, j(1D)1 = 2 3D j2(5) = 1, 40 j1(7) = 0 &
j1(5) = - 1 &
j3(8) = 0,
j5(B) = 1 j2(B) = - 2 j2(8) = 4
41 j2(3) = 1 & 42 j1(2) = 0 &
j14(15) = 1 j20(21) = 1,
j9(15) = - 1, j10(21) = - 1,
j4(15) = - 2,
j9(21) = 3,
j(15)1 = - 4 j6(21) = 5, j(21)1 = - 2
46
43 j42(43) = 1 j21(43) = -1, jB(43) = -5, j(43)5 = - 3, j(43)3 = 2 44 j2(4) = 0 & 45 j6(45) = 1,
j2(11) = 1, j1(11) = - 1 j3(45) = - 1
46 j1(2) = 0 & j11(23) = 1, j9(23) = 4, j5(23) = 5, j(23)3 = 2 47 j1(7) = 0 & 48 j6(48) = 0,
j6(9) = 1 j3(9) = -1 j2(48) = 4
49 j2(5) = 1, j1(5) = - 1 & j1(D) = 1 4A j1(2) = 0 & j2(3) = 1 & j5(B) = 1, j2(B) = - 2 4B jB(4B) = 1, 4C j2(4) = 0 &
j6(4B) = 9, j12(13) = 1,
j3(4B) = - 3 j8(13) = - 1, j(13)2 = 2
4D j2(3) = 1 & 50 j1(2) = 0 &
j18(19) = 1, j1(7) = 0 &
jB(19) = -1, j2(5) = 1,
j6(19) = 3, j(19)2 = 2 j1(5) = - 1
51 jA(51) = 1, j5(51) = - 1, j(51)1 = - 5 52 j3(8) = 0, j2(8) = 4 & j6(9) = 1 j3(9) = -1 53 j52(53) = 1, j28(53) = - 1, j12(53) = 2, j(53)6 = - 3, j(53)4 = - 4 54 j1(2) = 0 & 55 j2(3) = 1 & 56 j2(4) = 0 &
jC(29) = 1 jA(1B) = 1, j14(15) = 1
j6(29) = - 1, j5(1B) = - 1,
j9(15) = - 1,
j3(29) = 6 j2(1B) = - 4
j4(15) = - 2, j(15)1 = - 4
57 j1(7) = 0 & j5(B) = 1, j2(B) = - 2 58 j1(2) = 0 & 59 j1C(59) = 1,
j2(3) = 1 & jD(59) = - 1,
j1(D) = 1
j5(59) = - 8, j(59)3 = 11 5A j2(5) = 1 & j4(12) = 0, j2(12) = 4 5B j3C(5B) = 1, j1D(5B) = - 1, j8(5B) = - 2, j(5B)3 = 8 5C j1(2) = 0 & j8(2D) = 1, j4(2D) = - 1, j(2D)1 = 3
47
5D j5C(5D) = 1 j2D(5D) = - 1, jB(5D) = 11, j8(5D) = 3, j(5D)9 = 2, j3(5D) = 5, 60 j2(4) = 0 & j1(7) = 0 & j2(3) = 1 61 j2(5) = 1, 62 j1(2) = 0 &
j1(5) = - 1 &
j17(31) = 1
j12(13) = 1, j11(31) = - 2,
j8(13) = - 1,
j9(31) = 4,
j(13)2 = 2 j(31)4 = - 5, j(31)1 = - 3
63 j2(3) = 1 & j20(21) = 1, j10(21) = - 1, j9(21) = 3, j6(21) = 5, j(21)1 = - 2 64 j3(8) = 0, j2(8) = 4 & j5(B) = 1, j2(B) = - 2 65 j64(65) = 1 j32(65) = - 1, j(65)8 = 2, j5(65) = - 3 66 j1(2) = 0 & j2(5) = 1, j1(5) = - 1 & j6(9) = 1 j3(9) = -1 67 j1(7) = 0 & j1(D) = 1 68 j2(4) = 0 & j18(19) = 1, jB(19) = -1, j6(19) = 3, j(19)2 = 2 69 j2(3) = 1 & 6A j1(2) = 0 &
j11(23) = 1, j19(35) = 1,
j9(23) = 4, j9(35) = 4,
j5(23) = 5, j5(35) = 3,
j(23)3 = 2 j(35)7 = 2, j2(35) = 8
6B j2(5) = 1, j1(5) = - 1 & j14(15) = 1 j9(15) = - 1, j4(15) = - 2, j(15)1 = - 4 6C j2(3) = 1 & j5(24) = 0, j2(24) = 4 6D j6C(6D) = 1, j36(6D) = - 1, j13(6D) = - 5, j(6D)A = - 3, j4(6D) = 4, j2(6D) = 2 70 j1(2) = 0 & j2(37) = 0 71 j6(9) = 1 j3(9) = -1 & j5(B) = 1, j2(B) = - 2 72 j2(4) = 0 & jA(1B) = 1, j5(1B) = - 1, j2(1B) = - 4
TABLE 9 (BASE- 15)
Here the first 10 digits are the same digits of the decimal system – 0,1,2,3,4,5,6,7,8 and 9.The remaining five
digits are the symbol A,B,C,D and E representing decimal values 10,11,12,13 and 14. Each position in this number
system represents a power of the base (15)
Divisor d
Divisibility rule/test Jg(d) = m / J(d)g = m
2 j1(2) = 1
48
3 j1(3) = 0 4 j2(4) = 1, 5 j1(5) = 0 6 j1(3) = 0 & 7 j1(7) = 1 8 j2(8) = 1, 9 j2(9) = 0
j1(4) = - 1
j1(2) = 1
j1(8) = - 1
A j1(2) = 1 & B j5(B) = 1, C j1(3) = 0 & D jC(D) = 1,
j1(5) = 0 j(B)1 = 3 j2(4) = 1,
j6(D) = - 1,
j1(4) = - 1 j(D)2 = - 3
E j1(E) = 1 10 j1(10) = 0 11 j2(11) = 1 12 j8(12) = 1, 13 j2(9) = 0 & 14 j13(14) = 1,
j4(12) = - 1, j1(2) = 1 j9(14) = - 1,
j2(12) = 4
j5(14) = 2, j(14)1 = - 5
15 j1(5) = 0 & 16 j1(3) = 0 & 17 j1(2) = 1 &
j2(4) = 1, j1(7) = 1 j5(B) = 1,
j1(4) = - 1
j(B)1 = 3
18 j17(18) = 1, jB(18) = - 1, j4(18) = 2, j(18)1 = - 3 19 j1(3) = 0 & j2(8) = 1, j1(8) = - 1 1A j2(1A) = 0 1B j1(2) = 1 & jC(D) = 1, j6(D) = - 1, j(D)2 = - 3
49
1C j3(1C) = 0 1D j2(4) = 1, j1(4) = - 1 & j1(7) = 1 1E j1D(1E) = 1, jE(1E) = -1, j9(1E) = -3, j(1E)2 = 4, j(1E)1 = 2 20 j1(10) = 0 & 21 jA(21) = 1,
j1(2) = 1 j5(21) = -1, j(21)1 = - 2
22 j2(22) = 1 23 j1(3) = 0 & 24 j1(2) = 1 & 25 j1(5) = 0 & 26 j2(9) = 0 &
j5(B) = 1, j8(12) = 1, j1(7) = 1 j2(4) = 1,
j(B)1 = 3
j4(12) = - 1, j2(12) = 4
j1(4) = - 1
27 j26(27) = 1, j13(27) = - 1, j7(27) = - 2, j(27)4 = - 4, j2(27) = 3
28 j1(2) = 1 & j13(14) = 1, j9(14) = - 1, j5(14) = 2, j(14)1 = - 5 29 j1(3) = 0 & 2A j1(5) = 0 &
jC(D) = 1, j2(8) = 1,
j6(D) = - 1, j(D)2 = - 3 j1(8) = - 1
2B j2A(2B) = 1, j15(2B) = -1, j10(2B) = - 3, j6(2B) = 5, j(2B)2 = - 2 2C j1(3) = 0 & j1(E) = 1 2D j16(2D) = 1, j10(2D) = 4, j5(2D) = -5, j(2D)3 = - 2 2E j2(4) = 1, j1(4) = - 1 & j5(B) = 1, j(B)1 = 3 30 j2(30) = 0 31 j1(2) = 1 & 32 j31(32) = 1,
j17(18) = 1, j18(32) = - 1,
jB(18) = - 1, jE(32) = -2,
j4(18) = 2, j5(32) = - 4,
j(18)1 = - 3 j4(32) = 6
33 j1(3) = 0 & 34 j7(34) = 1,
j2(11) = 1 j3(34) = - 6
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35 j1(2) = 1 & j2(1A) = 0 36 j1(3) = 0 & j8(12) = 1, j4(12) = - 1, j2(12) = 4 37 j2(4) = 1, j1(4) = - 1 & jC(D) = 1, j6(D) = - 1, j(D)2 = - 3 38 jD(38) = 1, 39 j1(2) = 1 & 3A j1(5) = 0 & 3B j1(7) = 1 &
j(38)2 = - 4 j3(1C) = 0 j5(B) = 1, j2(8) = 1,
j(B)1 = 3 j1(8) = - 1
3C j1(3) = 0 & 3D j1(2) = 1 &
j13(14) = 1, j1D(1E) = 1,
j9(14) = - 1, jE(1E) = -1,
j5(14) = 2, j9(1E) = -3,
j(14)1 = - 5 j(1E)2 = 4, j(1E)1 = 2
3E j1E(3E) = 1, je(3E) = - 2, j4(3E) = 3, j(3E)1 = 4 40 j2(4) = 1, 41 j10(41) = 1 42 j1(2) = 1 &
j1(4) = - 1 &
j(41)3 = - 3,
jA(21) = 1,
j1(10) = 0 j(41)1 = - 4
j5(21) = -1, j(21)1 = - 2
43 j1(7) = 1 & 44 j4(44) = 1
j2(9) = 0 j(44)1 = - 12
45 j1(5) = 0 & 46 j1(3) = 0 &
jC(D) = 1, j1(2) = 1 &
j6(D) = - 1,
j5(B) = 1,
j(D)2 = - 3 j(B)1 = 3
47 jB(47) = 1, j5(47) = - 3, j(47) 4= - 5 48 j2(4) = 1, j1(4) = - 1 & j8(12) = 1, j4(12) = - 1, j2(12) = 4 49 j1(3) = 0 & j17(18) = 1, jB(18) = - 1, j4(18) = 2, j(18)1 = - 3 4A j1(5) = 0 & j1(E) = 1 4B j25(4B) = 1, j8(4B) = 4, j(4B)6 = 3, j4(4B) = 2 4C j2(9) = 0 & j2(8) = 1, j1(8) = - 1 4D j4C(4D) = 1, j26(4D) = - 1, j15(4D) = - 11, j6(4D) = - 3, j(4D)4 = - 2
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4E j1(2) = 1 & j26(27) = 1, j13(27) = - 1, j7(27) = - 2, j(27)4 = - 4, j2(27) = 3 50 j1(3) = 0 & j2(1A) = 0 51 j2(4) = 1, j1(4) = - 1 & j13(14) = 1, j9(14) = - 1, j5(14) = 2, j(14)1 = - 5 52 j1(7) = 1 & j5(B) = 1, j(B)1 = 3 53 j1(3) = 0 & j1(2) = 1 & jC(D) = 1, j6(D) = - 1, j(D)2 = - 3 54 j1B(54) = 1, jD(54) = - 1, j7(54) = - 8, j(54)4 = - 12 55 j1(5) = 0 & j2(11) = 1 56 j4(56) = 0 57 j1(2) = 1 & 58 j57(58) = 1,
j2A(2B) = 1, j2B(58) = - 1
j15(2B) = -1, j(58)c = - 2,
j10(2B) = - 3, j4(58) = - 5,
j6(2B) = 5, j(2B)2 = - 2 j(58)3 = - 3
59 j1(3) = 0 & 5A j1(5) = 0 &
j2(4) = 1, j8(12) = 1,
j1(4) = - 1 &
j4(12) = - 1,
j1(7) = 1 j2(12) = 4
5B j1(2) = 1 & j16(2D) = 1, j10(2D) = 4, j5(2D) = -5, j(2D)3 = - 2 5C j1(3) = 0 & j1D(1E) = 1, jE(1E) = -1, j9(1E) = -3, j(1E)2 = 4, j(1E)1 = 2 5D j2(8) = 1, j1(8) = - 1 & j5(B) = 1, j(B)1 = 3 5E j5D(5E) = 1, j2E(5E) = - 1, jC(5E) = - 2, j4(5E) = - 11 60 j2(30) = 0 & j1(2) = 1 61 j1(7) = 1 & jC(D) = 1, j6(D) = - 1, j(D)2 = - 3 62 j2(4) = 1, j1(4) = - 1 & j17(18) = 1, jB(18) = - 1, j4(18) = 2, j(18)1 = - 3 63 j1(3) = 0 & jA(21) = 1, j5(21) = -1, j(21)1 = - 2 64 j1(2) = 1 & j31(32) = 1, j18(32) = - 1, jE(32) = - 2, j5(32) = - 4, j4(32) = 6 65 j1(5) = 0 & j13(14) = 1, j9(14) = - 1, j5(14) = 2, j(14)1 = - 5 66 j1(3) = 0 & j2(22) = 1 67 j66(67) = 1, j33(67) = - 1, jE(67) = 2, j8(67) = -11, j4(67) = - 9
52
68 j1(2) = 1 & 69 j2(9) = 0 & 6A j2(1A) = 0 &
j7(34) = 1, j5(B) = 1, j2(4) = 1,
j3(34) = - 6 j(B)1 = 3 j1(4) = - 1
TABLE 10 (BASE -16)
Here the first 10 digits are the same digits of the decimal system – 0,1,2,3,4,5,6,7,8 and 9.The remaining six digit
are the symbol A,B,C,D,E and F representing decimal values 10,11,12,13,14 and 15. Each position in this number
system represents a power of the base (1610 = 1016 ).
Divisor d
Divisibility rule/test Jg(d) = m / J(d)g = m
2 j1(2) = 0 3 j1(3) = 1 4 j1(4) = 0 5 j1(5) = 1 6 j1(2) = 0 & 7 j3(7) = 1 8 j1(8) = 0 9 j3(9) = 1 A j1(2) = 0 & B j5(B) = 0 & C j1(4) = 0 & D j3(D) = 1 E j1(2) = 0 & F j1(F) = 1 10 j1(10) = 0
j1(3) = 1
j1(5) = 1 j2(B) = 3 j1(3) = 1
j3(7) = 1
53
11 j2(11) = 1, 12 j1(2) = 0 & 13 j9(13) = 0, 14 j1(4) = 0 & 15 j3(15) = 1
j1(11) = - 1 j3(9) = 1
j(3)2 = - 2 j1(5) = 1
16 j1(2) = 0 & 17 jB(17) = 1, 18 j1(8) = 0 &
j5(B) = 1, j2(B) = 3 j3(17) = 2 j1(3) = 1
19 j5(19) = 1, 1A j1(2) = 0 & 1B j9(1B) = 1, 1C j1(4) = 0 & 1D j7(1D) = 1, 1E j1(2) = 0 & 1F j5(1F) = 1, 20 j2(20) = 0 21 j5(21) = 1, 22 j1(2) = 0 & 23 j3(23) = 1 24 j1(4) = 0 & 25 j9(25) = 1, 26 j1(2) = 0 & 27 j3(27) = 1 28 j1(8) = 0 &
j(19)2 = - 4
j3(D) = 1 j(1B)2 = - 2
j3(7) = 1
j(1D)3 = - 4
j1(F) = 1
j(1F)1 = 2
j(21)1 = - 2
j2(11) = 1,
j3(9) = 1 j5(25) = - 4,
j9(13) = 1,
j1(5) = 1
j1(11) = - 1
j(25)1 = 7 j(13)2 = - 2
54
29 j5(29) = 1, 2A j1(2) = 0 & 2B j7(2B) = 1, 2C j1(4) = 0 &
j(29)2 = - 4 j3(15) = 1 j2(2B) = - 2 j5(B) = 1, j2(B) = 3
2D j3(2D) = 1 2E j1(2) = 0 & 2F j17(2F) = 1, 30 j1(10) = 0 & 31 j15(31) = 1, 32 j1(2) = 0 & 33 j1(3) = 1 &
jB(17) = 1, j6(2F) = 2,
j1(3) = 1 j10(31) = 2, j5(19) = 1, j2(11) = 1,
j3(17) = 2 j(2F)1 = 3
j(31)5 = 2, j(31)1 = - 3 j(19)2 = - 4 j1(11) = - 1
34 j1(4) = 0 & j3(D) = 1 35 jD(35) = 1, 36 j1(2) = 0 &
j7(35) = - 4, j9(1B) = 1,
j(35)6 = - 4, j(35)2 = - 6 j(1B)2 = - 2
37 j5(37) = 1, 38 j1(8) = 0 & 39 j9(39) = 1,
j1(5) = 1 &
j3(7) = 1 j5(39) = 4,
j2(B) = 3
j(39)2 = - 2 3A j1(2) = 0 & 3B j1D(3B) = 1,
j7(1D) = 1, jF(3B) = 4,
j4(1D) = - 4 jA(3B) = 11, j(3B)2 = 3
3C j1(4) = 0 & j1(F) = 1 3D jF(3D) = 1, j8(3D) = - 4, j(3D)6 = - 3, j(3D)2 = - 5 3E j1(2) = 0 & 3F j3(3F) = 1,
j5(1F) = 1, j(1F)1 = 2 j(3F)1 = 4
40 j2(40) = 0
55
41 j3(41) = 1, 42 j1(2) = 0 &
j2(41) = - 4, j5(21) = 1,
j(41)1 = - 4 j(21)1 = - 2
43 j21(43) = 1, j19(43) = - 2, j(43)8 = - 2, j4(43) = A 44 j1(4) = 0 & 45 jB(45) = 1,
j2(11) = 1 j6(45) = 4,
j1(11) = - 1 j1(3) = 1 & j3(17) = 2
46 j1(2) = 0 & 47 j23(47) = 1, 48 j1(8) = 0 & 49 j9(49) = 1, 4A j1(2) = 0 & 4B j1(3) = 1 & 4C j1(4) = 0 & 4D j3(7) = 1 &
j3(23) = 1 j9(47) = 2,
j3(9) = 1 j5(49) = 4, j9(25) = 1, j5(19) = 1,
j9(13) = 1 j5(B) = 1,
j4(47) = 3
j(49)2 = 2
j5(25) = - 4, j1(25) = 7
j(19)2 = - 4 j(13)2 = - 2 j2(B) = 3
4E j1(2) = 0 & j3(27) = 1 4F j27(4F) = 1, 50 j1(10) = 0 & 51 j1B(51) = 1, 52 j1(2) = 0 & 53 j29(53) = 1, 54 j1(4) = 0 &
jA(4F) = 2,
j1(5) = 1 j7(51) = -2, j5(29) = 1, j12(53) = 3, j3(15) = 1
j(4F)4 = -7, j(4F)1 = 5
j4(51) = 7 j(29)2 = - 4
j7(53) = A
55 j2(55) = 1 56 j1(2) = 0 & 57 j7(57) = 1, 58 j1(8) = 0 &
j7(2B) = 1, j3(57) = 7 j5(B) = 1,
j2(2B) = - 2
j2(B) = 3
56
59 jB(59) = 1, 5A j1(2) = 0 & 5B j3(5B) = 1 5C j1(4) = 0 & 5D j5(5D) = 1, 5E j1(2) = 0 &
j3(59) = 2 j3(2D) = 1
jB(17) = 1 j3(5D) = 4
j17(2F) = 1,
j3(17) = 2
j6(2F) = 2, j(2F)1 = 3
5F j9(5F) = 1, j(5F)1 = 6 60 j2(20) = 0 & 61 jC(61) = 1,
j1(3) = 1 j6(61) = - 1, j(61)1 = - 6
62 j1(2) = 0 & j15(31) = 1, j10(31) = 2, j(31)5 = 2, j(31)1 = - 3 63 j3(9) = 1& j5(B) = 1, j2(B) = 3 64 j1(4) = 0 & j5(19) = 1 j(19)2 = - 4
Finally, we present here some properties of the above mentioned rules/tests. Here d is co-prime to the base and
all other variables are integers.
1 If d is a prime number then Jd-1(d) = 1, and J(d-1)/2 (d) = - 1 2 If Jn(d) = 1, n is even and d is a prime number then Jn/2(d) = - 1, 3 If Jm(d) = a and Jn(d) = b then Jm+n(d) = ab (mod d) 4 If Jm(d) = 1, Jn(e) =1, g.c.d. (d,e) = 1 and l.c.m. (m,n) = p then Jp(d e) = 1 Acknowledgement: Our thanks are due to the manufacturers of 12-digit, 112 Steps Check & Correct, Cltllzen
Electronic Calculator, China.
V.&P.0. Shiv Pur, Distt. Sirmour (H.P.) 173 025 INDIA E-mail address:[email protected]
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