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**NUMBERS** :

1. Prime Number : A Prime Number is a number which has factors only unity and itself. Eg. 2,3,5,7…..

2. Composite Number : A number which has more than two different factors is called a Composite Number. Eg. 4,6,8,10…

3. Co-Prime Number : Two numbers are said to be Co-Prime to each other if they do not have any common factor other than one. Eg. (2,3), (5,7)….

4.Twin Prime Numbers : Two prime numbers are said to be twin primes if the difference between them is two. Eg. (3,5), (5,7), (11,13)….

5. Perfect Number : A number is said to be perfect number if the sum of all the positive integral divisors of the numbers including unity and itself is equal to twice the number.

Number = P1a1. P2a2 . P3a3 …. Pnan

Where P1, P2, P3….. Pn are n distinct primes

and ai, i = 1,2,3,…..n is the number of times then the n prime is repeated then we have to use the following formulae

i. The number of positive integral divisors of the number including unity and itself

= (a1+1) (a2+1) (a3+1)….(an+1)

ii. The sum of all the positive integral divisors of the number including unity and itself

6. 1. If ‘a’ is even, 2a -1 is always divisible by 3

2. If ‘a’ is odd, 2a -1 is always divisible by 3

3. If ‘a’ is odd, 22a -1 is always divisible by 5

4. If ‘a’ is even, 22a -1 is always divisible by 5

5. If ‘a’ is odd, 52a -1 is always divisible by 13

6. If ‘a’ is even, 52a -1 is always divisible by 13

7. Tests for Divisibility:

i. A number is divisible by 2, when its unit digit even or zero

Eg. 262,240

ii. A number is divisible by 3, when the sum of its digits is divisible by 3.

Eg. 156 , 27

iii. A number is divisible by 4, when the number formed by the last two right hand digits is divisible by 4 or if the last two digits are zero. Eg. 144,1200

iv. A number is divisible by 5, when its units digit is 5 or 0. Eg. 1245,1350

v. A number is divisible by 6, when it is divisible by 2 and 3 both.

Eg. 126, 78

vi. No rule is still known for divisibility of a number by 7.

vii. A number is divisible by 8, when the number formed by the last three right hand digits its divisible by 8 (or) when the last three digits are 0’s

Eg. 4408, 1200

viii. A number is divisible by 9, when the sum of the digits is divisible by 9.

Eg. 1926 , 1359

ix. A number is divisible by 10, when its unit digit is 0

Eg. 4000,1850

x. A number is divisible by 11, when the difference between the sum of the digits in the odd and even places is zero or a multiple of 11

Eg. 91531, 2332

xi. A number is divisible by 12, when it is divisible by 3 and 4 both.

Eg. 132, 2556

xii. A number is divisible by 25, when the number formed by the last two right hand digits is divisible by 25. Eg. 143575, 12150

xiii. A number is divisible by 125, when the number formed by the last three right hand digits is divisible by 125. Eg. 139625, 460250

xiv. For any integer ‘a’

a3 – a is divisible by 3

a5 – a is divisible by 5

a7 – a is divisible by 7

a11 – a is divisible by 11

a13 – a is divisible by 13

In general, if P is a prime number then for any whole number ‘a’

ap – a is divisible by P.

xv. When any number with even number of digits is added to its reverse,

the sum is always a multiple by 11.

Eg. 5762 + 2675 = 8437 which is divisible by 11

Note: A number is divisible by another number if the former is divisible by all the factors of the later.

11. Sum of first ‘n’ odd natural numbers = n^{2}

12. Sum of first ‘n’ odd natural numbers = n(n+1)

13. In division sum we have dividend, divisor, quotient and remainder.

Dividend = Divisor × Quotient + Remainder

(or)