What Are Prime Numbers? A Beginner’s Guide

Prime numbers are the building blocks every other whole number is made from, and the definition is simpler than it might seem, though one common exception (the number 1) trips people up regularly.

What Is a Prime Number?

A prime number is a whole number greater than 1 that has exactly two positive divisors: 1 and itself. Numbers with more than two divisors, like 12 (divisible by 1, 2, 3, 4, 6, and 12), are called composite numbers instead.

Why Isn’t 1 a Prime Number?

By the definition above, a prime number needs exactly two distinct divisors. The number 1 only has one divisor, itself, so it doesn’t meet the definition and is classified as neither prime nor composite.

How to Check If a Number Is Prime

The straightforward method is trial division: check whether the number is evenly divisible by any integer from 2 up to its square root. If none of them divide evenly, the number is prime.

Worked Example

Common Mistakes

• Assuming 1 is prime. It’s specifically excluded by the definition, since it only has one divisor rather than two.
• Checking divisibility all the way up to the number itself. Only checking up to the square root is necessary; if no divisor exists below that point, none exists above it either, since divisors pair up around the square root.
• Assuming all odd numbers are prime. Plenty of odd numbers are composite, like 9 (3 × 3) or 15 (3 × 5); being odd only rules out divisibility by 2, not by every other number.

Use the Prime Number Checker

The Prime Number Checker checks whether any number is prime instantly, without manual trial division.