# Is 2 prime?

This is part of a series on common misconceptions.

Is this true or false?

\[ \text{The number 2 is prime.}\]

**Why some people say it's true:** It is so small, it's prime.

**Why some people say it's false:** Even numbers are not prime.

The statement is \( \color{green}{\textbf{true}}\).

Proof:The definition of a prime number is a positive integer that has exactly two distinct divisors. Since the divisors of 2 are 1 and 2, there are exactly two distinct divisors, so 2 is prime.

Rebuttal: Because even numbers are composite, 2 is not a prime.

Reply: That is true only for all even numbers greater than 2. If a number is of the form \( n = 2k \) with \( k > 1 \), then we know it has the distinct factors 1, 2, and \( 2k \), and thus it cannot be prime. However, for \( k =1 \), we have \( 2 = 2k,\) so there are only 2 factors.

**See Also**