A number is divisible by the given integer if the condition is met. If the number is divisible by both 2 and 3. If this difference is divisible by 11, then the number is divisible by 11. 14 is a factor if both 2 and 7 are, but there isn't an easy test for 7. Every integer greater than one is either a prime number number or can be expressed as an unique product of prime numbers.
Prime factorization is the technique used to find that unique factorization.
Primes appear many times in arithmetic, hence the reason this is the fundamental theorem. When reducing fractions, it is easy to see what to reduce if the numerator and denominator have been written in terms of primes. Prime numbers can be used in the divisibility tests to arrive at divisibility tests for composite numbers. The index of a radical must be larger than the exponent on each prime factor in the radicand for a radical to be in simplified radical form. The set of real numbers is closed under addition and multiplication.