# Numbers

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## Basic formulas for Number System

### Prime Numbers:

A number 'n' is prime if it has only two factors '1' and 'n'. There are 25 prime numbers till 100. They are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97. 2 is the only even prime number. All other prime numbers are odd.

Two numbers are considered to be co-prime if their H.C.F is 1.
Example: H.C.F. of 16 and 25 is 1. Therefore, 16 and 25 are co-primes.

### Test to identify a prime number 'n':

Consider a number 'p' such that p is the square root of 'm' where 'm' is a perfect square immediately greater than 'n'. If 'n' is NOT divisible by any of the prime numbers less than 'p', then 'n' is a prime number.

Example: Say n = 97, then m = 100 (a perfect square immediately greater than 97). Hence, p = 10. Now check if 97 is divisible by any of the prime numbers less than 10 i.e. by 2, 3, 5, 7. As 97 is NOT divisible by any of the prime numbers less than 10. Hence, 97 is a prime number.

### TESTS OF DIVISIBILITY

1) Divisibility by 2: A number 'N' is divisible by 2 if the unit's digit of 'N' is 0 or is divisible by 2 (0/2/4/6/8)
Example: 240, 256
Number 240 is divisible by 2 because the unit's digit of 240 is 0.
Number 256 is divisible by 2 because the unit's digit of 256 is 6 which is divisible by 2.

2) Divisibility by 4: A number 'N' is divisible by 4 if the last two digits of 'N' are 0's or are divisible by 4.
Example: 12400, 31648, 224 etc.
Number 12400 is divisible by 4 because the last two digits of 12400 are '00'.
Number 31648 is divisible by 4 because the last two digits of 31648 are divisible by 4.
Number 224 is divisible by 4 because the last two digits of 224 are divisible by 4.

3) Divisibility by 8: A number 'N' is divisible by 8 if the last three digits of 'N' are 0's or are divisible by 8.
Example: 12400, 31648, 11224, 21000 etc.
Number 12400 is divisible by 8 because the last three digits of 12400 are divisible by 8.
Number 31648 is divisible by 8 because the last three digits of 31648 are divisible by 8.
Number 11224 is divisible by 8 because the last three digits of 11224 are divisible by 8.
Number 21000 is divisible by 8 because the last three digits of 21000 are 0's.