Prime & Composite Numbers
Numbers with exactly two factors, and everything else.
Counting the factors
A prime number has exactly two factors: 1 and itself.
A composite number has more than two factors.
The primes you should know
Under 20 they are: 2, 3, 5, 7, 11, 13, 17, 19.
Learn these eight and most questions become quick.
Testing a number
Try dividing by 2, 3, 5 and 7 in turn. If none divides exactly, it is prime.
Is 21 prime? 21 ÷ 3 = 7 exactly, so no, it is composite.
Two special cases
1 is not prime. It has only one factor (itself) and a prime needs exactly two.
2 is the only even prime. Every other even number has 2 as an extra factor.
Finding primes with a sieve
Write 1 to 100. Cross out 1. Keep 2, cross out every other multiple of 2. Keep 3, cross out its multiples. Then 5, then 7.
What survives is the list of primes. This is the Sieve of Eratosthenes.
Why primes matter
Every composite number can be broken down into primes multiplied together.
12 = 2 × 2 × 3. Primes are the building blocks of all the other numbers.
Where marks go
- "1 is prime." It is not. It has only one factor.
- "All odd numbers are prime." 9, 15 and 21 are all odd and composite.
- "2 cannot be prime because it is even." It is prime, and the only even one.
What to remember
- Prime: exactly 2 factors. Composite: more than 2.
- Primes under 20: 2, 3, 5, 7, 11, 13, 17, 19.
- 1 is not prime; 2 is the only even prime.
Watch one worked through
Here is one done slowly, with the reason for every move. Then one for you.
Is 51 prime or composite?
- prime = exactly 2 factorsRecall what you are testing: a prime has exactly two factors, 1 and itself.Stating the test first stops the usual slip of calling any odd number prime. 51 is odd, and odd has nothing to do with it.
- 5 + 1 = 6, 6 ÷ 3 = 2 ✓Try 2. 51 is odd, so 2 is not a factor. Try 3: the digits 5 and 1 add to 6, and 6 divides by 3.The digit sum test for 3 is quick and works on any size of number. It is the fastest way past the number most people wrongly call prime.
- 51 ÷ 3 = 17So 3 divides 51. Work out 51 ÷ 3 = 17.Finding one factor other than 1 and itself is enough. You do not have to find all of them to answer the question.
- 51 = 3 × 17 → composite51 = 3 × 17, so it has more than two factors. It is composite.Only testing up to the square root is needed, which for 51 means up to 7. Beyond that any factor would already have shown up as a partner of a smaller one.
Answer: Composite, because 51 = 3 × 17
Is 57 prime or composite? Answer with one word.
Show the answer
composite
Practise Prime & Composite Numbers
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