Some numbers have many factor pairs and some have only one. That difference sorts the counting numbers into primes and composites.
Key idea
A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11, 13, …). A composite number has more than two factors (4, 6, 8, 9, 10, …). The number 1 has only one factor, so it is neither.
Why it works
Think of rectangles again. 7 squares make only one rectangle, a 1-by-7 row, so 7 is prime. 9 squares can also make a 3-by-3 square, so 9 is composite. A number is composite as soon as you find any factor pair besides 1 and itself.
Worked example: An odd composite
Is 51 prime or composite?
- 51 is odd, so 2 is not a factor. Try 3: $3 \times 17 = 51$.
- 51 has the factors 1, 3, 17 and 51: more than two.
Answer: Composite
Worked example: A prime
Is 53 prime or composite?
- Try 2, 3, 5 and 7: none divides 53 evenly.
- You can stop at 7, because $8 \times 8 = 64$ is already more than 53. So the only factors are 1 and 53.
Answer: Prime
Common mistake
Calling 1 prime, or thinking every odd number is prime. 1 has just one factor, and odd numbers like 9, 15, 21, 51 and 91 are composite. Test small factors (3, 5, 7) before deciding.