What Are Prime Numbers?

The Definition

A prime number is a whole number greater than 1 that can only be divided evenly by 1 and itself. That means no other number goes into it without leaving a remainder. The number 7 is prime because the only way to divide it evenly is 7 ÷ 1 = 7 or 7 ÷ 7 = 1. No other number works.

A number that isn't prime (meaning it CAN be divided by other numbers) is called a composite number. The number 12 is composite because it can be divided by 1, 2, 3, 4, 6, and 12.

Note: the number 1 is neither prime nor composite — it's in its own special category.

The First 25 Prime Numbers

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Notice anything? The number 2 is the only even prime number. Every other even number can be divided by 2, so they're all composite. After 2, all primes are odd — but not all odd numbers are prime (9, 15, 21, 25 are all odd but not prime).

How to Find Prime Numbers: The Sieve of Eratosthenes

About 2,200 years ago, a Greek mathematician named Eratosthenes invented a brilliant method for finding all prime numbers up to any limit. Here's how it works:

• Write down all numbers from 2 to your limit (say, 100)
• Circle 2 (the first prime) and cross out every multiple of 2 (4, 6, 8, 10, …)
• Circle 3 (the next uncrossed number) and cross out every multiple of 3 (6, 9, 12, 15, …)
• Circle 5 and cross out every multiple of 5
• Continue this process — every number that hasn't been crossed out is prime

It's called a "sieve" because you're filtering out composite numbers, and the primes are what remain — like using a strainer to catch gold nuggets.

Why Primes Matter

Prime numbers are the building blocks of all numbers. Every whole number greater than 1 can be written as a product of prime numbers in exactly one way. This is called the Fundamental Theorem of Arithmetic. For example: 12 = 2 × 2 × 3. The number 60 = 2 × 2 × 3 × 5. This "prime factorization" is unique to each number, like a fingerprint.

Primes also power modern internet security. When you buy something online, your credit card number is protected by encryption algorithms that rely on the fact that multiplying two large prime numbers is easy, but figuring out which two primes were multiplied together is extraordinarily difficult — even for the world's fastest computers.

Why Prime Numbers Matter

Prime numbers are far more than a classroom curiosity — they are the building blocks of all whole numbers. Just as every molecule is made of atoms, every whole number greater than 1 is either prime or can be broken into a unique combination of primes. The number 60, for example, is 2 × 2 × 3 × 5. This is called the Fundamental Theorem of Arithmetic, and it means primes are the atoms of the number world.

In the modern world, prime numbers power the encryption that keeps your online banking, messages, and passwords safe. RSA encryption — the system that protects most internet transactions — works because multiplying two large primes is easy, but figuring out which two primes were multiplied is extraordinarily difficult for computers. Without prime numbers, there would be no online security.

Where Kids Get Stuck

The most common confusion is about the number 1. Children often assume 1 is prime because it seems to fit the rule: "a number that can only be divided by 1 and itself." But mathematicians exclude 1 from the prime list because including it would break the Fundamental Theorem of Arithmetic — every number would have infinitely many prime factorizations (since you could keep multiplying by 1). It's a subtle but important distinction.

Another difficulty is testing whether larger numbers are prime. Children can easily check whether 7 or 13 is prime, but what about 91? It looks prime, but it's actually 7 × 13. The trick is to check divisibility only by primes up to the square root of the number. For 91, the square root is about 9.5, so you only need to test 2, 3, 5, and 7. Since 91 ÷ 7 = 13, it's composite.

Students also sometimes confuse prime with odd. All primes except 2 are odd, but plenty of odd numbers are not prime (9, 15, 21, 25, 27…). The number 2 is special: it's the only even prime and the smallest prime.

The Sieve of Eratosthenes

The most elegant way to find primes was invented over 2,200 years ago by a Greek mathematician named Eratosthenes. His method, called a "sieve," works like this:

• Write all numbers from 2 to 100 in a grid.
• Circle 2, then cross out every multiple of 2 (4, 6, 8, 10…).
• Circle the next uncrossed number (3), then cross out every multiple of 3.
• Continue with 5, then 7. After 7, every composite number up to 100 has already been crossed out.
• The circled numbers that remain are all the primes up to 100. There are exactly 25 of them.

This activity is perfect for children because it reveals prime numbers through pattern recognition rather than memorization. Watching the sieve filter out composite numbers makes the concept tangible and satisfying.

Try This at Home

• Sieve on paper — Print a 10×10 grid numbered 1–100 and walk through the sieve together. Use different colors for each prime's multiples.
• Factor trees — Pick a number like 72 and build a factor tree, breaking it down until every branch ends in a prime.
• Prime or not? — Take turns naming a number. The other person has to determine if it's prime and explain why.
• Largest prime hunt — Look up the current largest known prime number together and marvel at its size (it has millions of digits).

🔎 Explore the Prime Sieve Tool

Last reviewed: May 2026