What Are Prime Numbers and How Do You Find Them?
Understand what makes a number prime in plain English, two methods to check if any number is prime, and why prime numbers matter outside of math class.
Prime Numbers in Plain English
A prime number is a whole number greater than 1 that can only be divided evenly by 1 and itself. No other whole number divides into it without leaving a remainder.
Examples of prime numbers: 2 is prime because only 1 and 2 divide into it evenly. 7 is prime because only 1 and 7 work. 13 is prime. 17 is prime. Examples of numbers that are NOT prime: 4 can be divided by 1, 2, and 4. 9 can be divided by 1, 3, and 9. 15 can be divided by 1, 3, 5, and 15.
Two is the only even prime number. Every other even number can be divided by 2, which means it has at least three divisors (1, 2, and itself), so it is not prime. This makes 2 unique among primes.
How to Check If a Number Is Prime
The basic method: try dividing the number by every whole number from 2 up to the square root of the number. If none of them divide evenly (with zero remainder), the number is prime. If any do divide evenly, the number is not prime.
You only need to test up to the square root because if a number has a factor larger than its square root, it must also have a corresponding factor smaller than the square root. So you cover all possible factors by checking only up to that point.
Example: is 37 prime? The square root of 37 is approximately 6.08. Check 2: 37 divided by 2 is 18.5 — no. Check 3: 37 divided by 3 is 12.33 — no. Check 4: 37 divided by 4 is 9.25 — no. Check 5: 37 divided by 5 is 7.4 — no. Check 6: 37 divided by 6 is 6.17 — no. Since no whole number up to 6 divides 37 evenly, 37 is prime.
The Sieve of Eratosthenes (Simple Version)
If you want to find all prime numbers up to a certain limit, the Sieve of Eratosthenes is the most efficient manual method. It was invented by a Greek mathematician around 240 BC and still works perfectly today.
Write all whole numbers from 2 to your upper limit. Start at 2. Circle it (it is prime). Cross out every multiple of 2 that follows: 4, 6, 8, 10, and so on. Move to the next uncircled number — 3. Circle it. Cross out every multiple of 3 that is not already crossed out: 9, 15, 21, and so on. Continue this process. Each uncircled number you reach is prime. Circle it and cross out its multiples.
What remains uncrossed and circled after you complete the sieve are all the prime numbers up to your limit. The method feels tedious by hand but produces perfect results. For numbers up to 100, you only need to sieve primes up to 10 (the square root of 100).
Why Prime Numbers Matter Outside of Math Class
Prime numbers are the foundation of modern internet security and encryption. When you see a padlock icon in your browser indicating a secure connection, the security depends on a mathematical principle: it is very easy to multiply two large prime numbers together, but computationally very hard to factor the result back into its two original primes.
RSA encryption, the most widely used public-key cryptography system, works by multiplying two very large primes (each hundreds of digits long) to create a public key. Cracking the encryption requires factoring that enormous product back into its prime components — a task that would take classical computers longer than the age of the universe for sufficiently large primes.
Every whole number greater than 1 is either prime or can be expressed as a unique product of prime numbers. This is called the Fundamental Theorem of Arithmetic, and it is why mathematicians describe primes as the building blocks or atoms of all numbers. Understanding primes is foundational to number theory, cryptography, and computer science.