Arithmetic & Numbers · Grades 5–9
Prime Numbers Explained: Test, Factor and Count Divisors
Primes and composites
2, 3, 5, 7, 11 and 13 are prime. 1 is not prime (it has one divisor), and 2 is the only even prime. A number with more than two divisors is composite.
Factor a composite number
Divide by 2 while you can, then 3, then 5, and so on. The primes you collect are the prime factorization: 84 = 2² × 3 × 7.
84
- Divisors: 12
- Sum of divisors: 224
Step-by-step working 3 steps
- 1Divide by the smallest prime that fits, again and again
84 ÷ 2 = 42 42 ÷ 2 = 21 21 ÷ 3 = 7 7 ÷ 7 = 1
Stop when the quotient reaches 1.
- 2Collect the primes
84 = 2 × 2 × 3 × 7 84 = 2² × 3 × 7
- 3Count the divisors
(3 × 2 × 2) = 12 divisors
Add 1 to each exponent and multiply.
Test for a prime
Test divisibility by primes up to the square root. If none divide the number, it is prime: 97 is prime.
97
- Divisors: 2
- Sum of divisors: 98
Step-by-step working 3 steps
- 1Divide by the smallest prime that fits, again and again
97 ÷ 97 = 1
Stop when the quotient reaches 1.
- 2Check
Only 97 itself divides 97 without a remainder (besides 1), so 97 is prime.
- 3Count the divisors
(2) = 2 divisors
Add 1 to each exponent and multiply.
A number that only looks prime
1001 has no obvious small factors, but it equals 1001 = 7 × 11 × 13.
Common mistakes
Treating 1 as prime
Primes need exactly two divisors.
Stopping at a composite factor
Keep factoring until every factor is prime.
Testing too many divisors
You never need to test beyond the square root.
Frequently asked questions
Are there infinitely many primes?
Yes. Euclid proved it over two thousand years ago.
How many divisors does a number have?
Add 1 to each exponent in its prime factorization and multiply.
