solutions 1

MATH 100 - WORKSHEET SOLUTIONS

1. Worksheet 1A

(1) What numbers are congruent to 0 mod 4?

. . . ,8,4, 0, 4, 8, . . .(2) What numbers are congruent to 1 mod 4?



. . . ,5,1, 3, 7, 11, . . .(5) 11 mod 4 = 3(6) 14 mod 4 = 2(7) 27 mod 4 = 3(8) 20 mod 4 = 0(9) 77 mod 4 = 1(10) 1 mod 4 = 3(11) 7 mod 4 = 1(12) 77 mod 4 = 3(13) Calculate in Z4.

(a) 1 + 2 = 3(b) 2 + 2 = 0(c) 2 + 3 = 1(d) 3 + 1 = 0(e) 0 2 = 0(f) 2 2 = 0(g) 2 3 = 2(h) 3 3 = 1

(14) Make the operation tables for Z4.

Addition

0 1 2 31 2 3 02 3 0 13 0 1 2

Multiplication0 0 0 00 1 2 30 2 0 20 3 2 1

(15) Solve these equations in Z4.

(a) x+ 2 = 3 x = 1(b) x+ 2 = 2 x = 0(c) x+ 2 = 1 x = 3(d) x+ 2 = 0 x = 2(e) x+ 3 = 1 x = 2(f) x+ 1 = 2 x = 1

Date: January 25, 2008.

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2 MATH 100 - WORKSHEET SOLUTIONS

(g) x+ 1 = 0 x = 3(h) x 1 = 2 x = 3(i) x 3 = 2 x = 1(j) x 1 = 3 x = 0

2. Worksheet 1B

(1) What numbers are congruent to 0 mod 5?




. . . ,12,5, 2, 9, 16, . . .(5) 11 mod 5 = 1(6) 14 mod 5 = 4(7) 27 mod 5 = 2(8) 20 mod 6 = 2(9) 77 mod 6 = 5(10) 1 mod 6 = 5(11) 7 mod 7 = 0(12) 9 mod 7 = 5(13) Calculate in Z7.

(a) 1 + 2 = 3(b) 4 + 6 = 3(c) 5 + 4 = 2(d) 2 + 5 = 0(e) 2 3 = 6(f) 2 4 = 1(g) 3 5 = 1(h) 4 6 = 3

(14) Make the operation tables for Z5.

Addition

0 1 2 3 41 2 3 4 02 3 4 0 13 4 0 1 24 0 1 2 3

Multiplication0 0 0 0 00 1 2 3 40 2 4 1 30 3 1 4 20 4 3 2 1

(15) Solve these equations in Z5. (Problems (b), (i) and (j) have numbers outside

the range 0, 1, 2, 3, 4 in them. This was unintentional, but in the event,the numbers are interpreted modulo 5.)(a) x+ 2 = 3 x = 1(b) x+ 5 = 2 x = 2(c) x+ 2 = 1 x = 4(d) x+ 4 = 0 x = 1(e) x+ 3 = 1 x = 3(f) x+ 1 = 4 x = 3

MATH 100 - WORKSHEET SOLUTIONS 3

(g) x+ 1 = 0 x = 4(h) x 1 = 3 x = 4(i) x 5 = 3 x = 3(j) x 4 = 6 x = 0

solutions 1

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