GMAT Math

Hopefully this will be a quick one, but does anyone know if negative numbers can be prime?

I'm thinking yes, but can someone please confirm? Thanks!

Of course for the GMAT all primes are positive.

However, negative one satisfies the Fundamental Theorem and the most common definition of a prime.

Positive one is not considered prime because it is divisible be exactly one integer rather than two.

Negative one on the other hand is divisible by exactly two integers (+1 and -1). Thus negative one satisfies the definition of a prime.

If we accept negative one as a prime then negative 2 does not satisfy the definition of the prime. It is divisible by 2, -1, -2, and 1. Since all negative integers are divisible by -1, the example with negative 2 above would apply to all negative integers less than -1. Thus the only possible negative prime is negative one.

Since that is the case, the Fundamental Theorem is satisfied if we do not let the prime factor negative one have a power other than one. In other words we cannot factor 5 into 5 X -1^2.

Thus a negative number such as - 120 can be broken down into a unique set of prime factors: 5 X 3 X 2^3 X -1.

This definition adds to the usefulness of the fundamental theorem allowing it to describe not only positive but also negative integers.

So why aren't negative numbers prime? It is because primes belong to a set of integers called natural numbers. Natural numbers are counting numbers. The first natural number is 1. So the definition of a prime is that it must first be a positive integer and then must be divisible by two, and only two, positive integers.

As for historically not considering negatives to be prime, the definition of primes has changed over time. The number one was considered prime throughout most of history.