Simple Proof Methods

Certain numbers are divisible by many integerts, while others are divisible by only a couple positive integers. These latter numbers have a special name.

Definition. [Prime numbers] A prime number $p > 1$ is an integer that has exactly two positive divisors: $1$ and $p$ (itself).

Formally, an integer $p$ is prime if, for all positive integers $d$, the only solutions to $d \mid p$ are $d = 1$ and $d = p$.

For example, $2, 3, 5, 7, 11, \dots$ are all prime numbers.

Numbers that have more than two divisors are called composite numbers, such as $4, 6, 8, 9, 10, \dots$.

Bonus question: if $p$ and $q$ are both primes, when is $p + q$ an odd integer?