Prime Numbers

A prime number is a whole number bigger than $1$ that has exactly two factors, $1$ and itself.
No other number divides it evenly.
Numbers with more than two factors, like $6$ or $9$, are called composite.

The number $2$ is the only even prime, since every other even number is divisible by $2$.
The number $1$ is not prime, because it has only one factor, not two.
So the list of primes starts $2, 3, 5, 7, 11$.

When a problem asks for the smallest primes, start from the beginning of that list.
The smallest prime is $2$, then $3$, then $5$.
For two different primes chosen as small as possible, use $2$ and $3$.

Primes are the building blocks of numbers.
If a number is a multiple of a prime like $5$ and is built from primes, then $5$ must be one of them.
No other prime is a multiple of $5$, which is often the key to number-property questions.

Is $7$ a prime number?
Its only factors are $1$ and $7$.
Since it has exactly two factors, $7$ is prime.

Is $9$ a prime number?
Besides $1$ and $9$, it is also divisible by $3$.
Because it has more than two factors, $9$ is not prime — it is composite.

A problem needs two different prime numbers, as small as possible. Which should you choose?
The two smallest primes are $2$ and $3$.
So use $2$ and $3$.