# Prime numbers

> To a number be considered as a prime number, it must be greater than 1 and only divisible by 1 and itself.

> Every integer greater than 1 can be written uniquely as a prime or as the product of two or more primes (prime factorization) where the prime factors are written in order of nondecreasing size.

641 = 641
100 = 2 x 2 x 5 x 5 = 2^2 x 5^2
999 = 3 x 3 x 3 x 37 = 3^3 x 37

### Trial Division

> If n is a composite integer (no-prime), then n has a prime divisor less than or equal to √n.

Show that 101 is prime

**Solution**
√101 ~ 10, so 101 will be prime if none of the prime numbers below 10 can divide 101.
The only primes not exceeding 101 are 2, 3, 5, and 7. Because 101 is not divisible by 2, 3, 5, or 7 (the quotient of 101 and each of these integers is not an integer), it follows that 101 is prime.