Numbers and sequences
Prime & composite numbers
Every whole number above 1 is one of two kinds: it either breaks into smaller factors, or it stubbornly refuses to.
Factors first
A FACTOR of a number divides into it exactly, with no remainder. The factors of 12 are 1, 2, 3, 4, 6 and 12. The factors of 7 are just 1 and 7. Finding factors in pairs helps you catch them all: 12 = 1 × 12 = 2 × 6 = 3 × 4.
Prime numbers
A PRIME NUMBER has exactly TWO factors: 1 and itself. The primes up to 30 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 Notice 2 is on the list. It is the only EVEN prime — every other even number can be divided by 2, so it has a third factor.
Composite numbers
A COMPOSITE NUMBER has MORE than two factors. 12 is composite: it has six factors. 9 is composite: 1, 3 and 9 — three factors. Every whole number above 1 is either prime or composite. There is no third possibility.
The special case of 1
1 is NEITHER prime nor composite. It has only ONE factor — itself. A prime needs exactly two, and a composite needs more than two. 1 fits neither, so mathematicians leave it out of both groups. This catches people out constantly, so it is worth remembering separately.
How to test a number
To check whether a number is prime, try dividing it by the small primes in turn: 2, 3, 5, 7, … If any of them divides exactly, the number is composite. If none does, it is prime. You can stop once the primes you are testing get too big — for a number under 100, testing 2, 3, 5 and 7 is enough.
Worked examples
Is 17 prime?
- Try dividing by 2: no, 17 is odd.
- By 3: 17 ÷ 3 leaves a remainder.
- By 5, by 7: neither divides exactly.
- So its only factors are 1 and 17.
Answer: Yes — 17 is prime
Is 15 prime?
- Try 2: no. Try 3: 15 ÷ 3 = 5 exactly.
- So 15 has factors 1, 3, 5 and 15 — that is four.
Answer: No — 15 is composite
Why 2 is the only even prime
- Every even number divides by 2.
- So any even number bigger than 2 has 1, 2 and itself as factors — at least three.
- Only 2 itself has exactly two.
Answer: 2 is the only even prime number
Is 1 prime?
- Count the factors of 1. There is only one: 1.
- A prime needs exactly two factors.
- A composite needs more than two.
Answer: 1 is neither prime nor composite
Making a prime from digit cards
- From the cards 1, 2, 5, 9 make a 2-digit number.
- Try 19: it does not divide by 2, 3, 5 or 7.
- So its only factors are 1 and 19.
Answer: 19 is prime
Where you see this in real life
- Prime numbers keep online messages secret — codes rely on how hard big primes are to split up.
- You cannot arrange 7 chairs into equal rows except 1 × 7, because 7 is prime.
- 12 works well for eggs and hours because it is composite with many factors.
Key words
- Factor
- A number that divides into another exactly.
- Prime number
- A number with exactly two factors: 1 and itself.
- Composite number
- A number with more than two factors.
- Multiple
- The result of multiplying a number by a whole number.
- Remainder
- What is left over after dividing.
- Divisible
- Divides exactly, with no remainder.