Prime and composite numbers

Two kinds of whole numbers

Every whole number greater than 1 is either prime or composite.

A factor is a number that divides another number with no remainder. If a number has a factor other than 1 and itself, it’s composite.

The special cases

• 1 is neither prime nor composite. It has only one factor (itself).
• 2 is the only even prime. Every other even number is divisible by 2, so it has at least three factors.
• 0 is not prime or composite either.

Seeing primes with arrays

Try to arrange counters in a rectangle (an array) with more than one row:

• 12 counters: 2 × 6, 3 × 4 ✓ → composite
• 7 counters: only 1 × 7 ✓ → prime (just one long row)

Primes can only make a single row.

Testing whether a number is prime

Try dividing by the small primes in order: 2, 3, 5, 7, …

• If any divides exactly, the number is composite.
• You only need to test primes up to the square root of the number. For numbers under 100, testing 2, 3, 5 and 7 is enough.

Example: Is 91 prime?

• 91 ÷ 2: no (odd)
• 91 ÷ 3: 9 + 1 = 10, not a multiple of 3, so no
• 91 ÷ 5: doesn’t end in 0 or 5, no
• 91 ÷ 7 = 13 ✓

So 91 = 7 × 13, and it is composite. Many people think 91 is prime. It isn’t!

The Sieve of Eratosthenes

A Greek mathematician, Eratosthenes, invented a neat way to find primes:

1. Write the numbers 1 to 100.
2. Cross out 1.
3. Circle 2, then cross out every multiple of 2.
4. Circle 3, then cross out every multiple of 3.
5. Circle the next number not crossed out (5) and cross out its multiples.
6. Repeat with 7.
7. Every number left is prime.

Primes up to 50

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

There are 15 primes less than 50 and 25 less than 100.

Worked examples

Example 1: Is 9 prime or composite?

9 = 3 × 3, so its factors are 1, 3 and 9: composite.

Example 2: Is 29 prime?

Not divisible by 2, 3 or 5 (and 7 × 4 = 28, 7 × 5 = 35). Prime.

Example 3: Why is 2 prime?

Its only factors are 1 and 2.

Common mistakes

• Calling 1 a prime number. A prime needs exactly two factors.
• Thinking all odd numbers are prime. 9, 15, 21, 25 and 27 are composite.
• Thinking all primes are odd. 2 is prime.

Try it yourself

1. Is 21 prime or composite?
2. Is 31 prime or composite?
3. List the primes between 50 and 60.
4. Is 57 prime? (Hint: add its digits.)

Answers: 1) composite (3 × 7) 2) prime 3) 53, 59 4) No: 5 + 7 = 12, so 57 = 3 × 19.

Quick recap

• A prime number has exactly two factors; a composite number has more.
• 1 is neither; 2 is the only even prime.
• Test by dividing by small primes, or use the Sieve of Eratosthenes.