Topic 1

Prime, composite & prime factorization

A prime number has exactly two factors: 1 and itself. A composite number has more than two. Prime factorization breaks a number into prime numbers that multiply to make it.

Class slides 4–13

How to answer questions

To test a number, see whether a smaller whole number divides it exactly.
If the only factors are 1 and itself, call it prime. If there are others, call it composite.
For prime factorization, keep splitting into factors until every factor is prime. Multiply them to check.

Worked example

Factorize 60:

60 = 6 × 10 = 2 × 3 × 2 × 5

All four factors are prime, so 60 = 2² × 3 × 5. Check: 4 × 3 × 5 = 60.

How to prepare

Know the primes up to 20: 2, 3, 5, 7, 11, 13, 17, 19. Practise one factor tree and one repeated-division problem.

Watch out: 1 is neither prime nor composite, and it never belongs in a prime factorization.

Try it:

Is 21 prime or composite? Write its prime factorization.

I can do this

Show answer

21 is composite because 3 × 7 = 21. Its prime factorization is 3 × 7.

More practice · 3 questions

Try all three before you open the solutions.

Warm-up

Is 29 prime or composite? Explain.

Show solution

29 is prime. It is not divisible by 2, 3, or 5, the prime numbers up to √29. Its only factors are 1 and 29.

Practice

Write the prime factorization of 84.

Show solution

84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7. So 84 = 2² × 3 × 7.

Challenge

The class slides ask about 1020. Write 1020 as a product of primes.

Show solution

Divide by small primes: 1020 ÷ 2 = 510, ÷ 2 = 255, ÷ 3 = 85, ÷ 5 = 17. So 1020 = 2² × 3 × 5 × 17.