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NUMBER THEORY PART - I

• Maximum Power Of a Number Dividing a Given Factorial

• Factors

• Congruent Modulo N

• Base System

• Cyclicity (Unit digit of a number)

• Congruent Modulo N

Maximum Power of a number dividing a given factorial

Finds the highest power of 5 that can divide 60!

60! …A huge number.

Impossible without

calculators

Solving with common sense

We know 60! = 1 x 2 x 3 x 4……..59 x 60

This contains

5 10 15 25 30 35….60

Therefore 14 is the answer

Total 12 in no. Obviously

. 512 can divide the

60!

But 25 & 50 are divisible by 5 twice. So power

goes up by two.

25

Alternate Method

60

5

12

5

25

0Adding we

get 14

Therefore 14 is the answer

Highest power of 5 that can divide 60!

A more complicated one

Find the highest power of 4 that can divide 24!

24

4

64

1

Adding we get 7

So the answer should be 7

BUT THIS IS INCORRECT

Explanation

24! = 1 x 2 x 3 … 24 contains

4 8 12 16 20 24 ( 6 in numbers)

As 16 is divisible by 4 twice therefore we get

7 as an answer

BUTBUT24! contains 2 & 6 also which are not divisible by 4But 2 x 6 is divisible by 4.Similarly 10 x 14 , 18 x 22 etc

How to do?

Find the highest power of 22 that can divide 24!

24

2

12

2

62

32

1

Adding we get

22

Earlier,the divisor given was prime number but 4 is not a prime number 4 can be written as 22

222 can divide 24! So (4)11 can divide 24!

75

5

15

5

3

Number of zeros in n!

# Find the number of zeros in the end of 75!OrOr

# Find the highest power of 10 that can divide 75!

FundaFundaNumbers of zeros depend upon number of 5’s and 2’s (10 = 2 X 5). So calculate maximum power of 5 dividing 75!

Adding we get 1818

So 75! has 1818 zeros at the end.

Factorization Trees

If a number n is not prime, we must be able to break it down to a product of prime numbers. Here is how,

60

6 10

2 3 2 5

60

2 30

2 3

5 6

However, the collection of prime numbers we get from the “leaves” of the tree is always the same. In other words, 60 = 2 × 2 × 3 × 560 = 2 × 2 × 3 × 5 no matter how we factorize it.

OrOr

Factors

Factors of 24

24 = 1 • 24 2 • 12 3 • 8 4 • 6

So the list of all

factors of 24 is

1, 2, 3, 4, 6, 8, 12, 24

Factors of 37

37 = 1 • 37 = 37 • 1

So the list of all the

factors is

1, 37

Factors of 64

64 = 1 • 64 2 • 32 4 • 16 8 • 8

So the list of factors

of 64 is

1, 2, 4, 8, 16, 32, 64

8 is only listed once

8 is only listed once

Numbers of Factors

Let us consider a number XX which can be written as XX = pa qb rc

Where p, q, and r are prime factor of the number and a, b, and c are Natural number.

Number of Factors = (a + 1)(b + 1)(c (a + 1)(b + 1)(c + 1)+ 1)

ExampleExample# # Find the numbers of factors of 24

Solution We can write 24 as 24 = 23.31

Numbers of Factors = (3 + 1)(1 + 1) = 4 x 2 = 88

Find the sum of the factors of 24.

The sum of the factors of a number can be found by using the prime factored form of the number.

13 3224

3210 2,2,2,2

To do this, use the prime factors themselves. Write the powers of each of the prime factors beginning with 0 and going to the power of the factor in the prime factored form.

10 3,3andThe sum of these are formed for each of the prime factors and then the product of these sums in found. )33)(2222( 103210

)31)(8421( )4)(15( 60

Wilson’s Theorem

If nn is a prime number, (n – 1)! + 1(n – 1)! + 1 is divisible by n.n.

Let’s take, n = 5 Then (n – 1)! + 1(n – 1)! + 1 = 4! + 1 = 24 + 1 = 25, which is divisible by 5.

ExampleExample

SimilarlySimilarly if n = 7 (n – 1)! + 1 = 6! + 1 = 720 + 1 = 721 which is divisible by 7.

If p is a prime number and N is prime to p, then Np-1 – 1 is divisible by p.

Example:

Take p = 3, N = 5 (3 and 5 are co-prime)

So, 53-1 – 1 = 24 is divisible by 3.

Fermat’s Theorem

Base System

What we are doing up to now we were doing on decimal base system. But other than this we have many other base system depending upon the number of initial digits used in, as in decimal base we use 10 digits (0,1,2,…..9).

Types of Base SystemTypes of Base System

(1) BinaryBinary : It consist of only two digits 0 & 1.

(2) OctalOctal : It has only 8 digits in it 0, 1, 2, 3, 4, 5, 6 & 7.

(3) DecimalDecimal: It is the commonly used base having 10 initial digits as 0, 1, 2, 3, 4, 5, 6, 7, 8 & 9.

(4) HexadecimalHexadecimal: It has 16 digits in it. these digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F.

Base SystemBase SystemThere can be following three types of questions

• Converting from any base to decimal baseConverting from any base to decimal base

• Converting from decimal to any baseConverting from decimal to any base

• Converting from one base to another (Other than Converting from one base to another (Other than decimal base system) decimal base system)

Convert abcd10 from decimal to N base.

= (pqrs)(pqrs)NN

p - q

abcd

Q1 - s

Q2 - r

N

N

NN

Base SystemBase SystemConverting from decimal to any baseConverting from decimal to any base

= (10352)(10352)88

1 - 0

8 - 3

4330

541 - 2

67 - 5

8

8

88

Convert 433010 from decimal to base 8.

ExampleExample

Base SystemBase SystemConverting from any base to decimal baseConverting from any base to decimal base

Let’s take the number as (abcd)N where a,b,c & d are the different digits and N is the base.

Number in decimal system = axN3 + bxN2 + cxN1 + dxN0

ExampleExample

What is the decimal equivalent of the number (2134)5?

Number in decimal system = 2x53 + 1x52 + 3x51 + 4x50

= 250 + 25 + 15 + 4 = 294

Base SystemBase SystemConverting from one base to anotherConverting from one base to another

First base

Decimal

Second base

Example

• Convert the number 1982 from base 10 to base 12. The result is

a. 1182 b. 1912 c. 1192 d. 1292

(CAT 1999)

Any Question??

Thank you