Prime Factors
Every question with a fully worked solution.
- Prime Factors - Video - Prime Factorization (Intro and Factor Trees) Watch
Theory
Factors are the numbers multiplied to get another number. Every whole number greater than \(1\) can be written as a product of prime factors: \(90 = 2 \times 3 \times 3 \times 5 = 2 \times 3^2 \times 5\). This Year 7 maths page shows factor trees, repeated division, index form and how prime factors find square roots.
A product of prime factors uses only prime numbers. \(1\) is not prime, so it is never included.
Factor tree. Split the number into a factor pair, then keep splitting until every branch ends in a prime. Circle the primes and collect them, smallest first. Whatever factor pair you start with, you always get the same primes.
Repeated division. Divide by the smallest prime that works. Keep going until you reach \(1\). The primes you divided by are the prime factors.
Index form. The index counts how many times each prime is used: \(2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2\). Check by multiplying back.
Square roots from prime factors. Split the primes into two equal groups; one group is the square root: \(441 = (3 \times 7) \times (3 \times 7)\), so \(\sqrt{441} = 21\). A number is a square number when every prime appears an even number of times.
Repeated division for 90:
| Divide by | Result |
|---|---|
| \(2\) | \(90 \div 2 = 45\) |
| \(3\) | \(45 \div 3 = 15\) |
| \(3\) | \(15 \div 3 = 5\) |
| \(5\) | \(5 \div 5 = 1\) |
How to write a number as a product of prime factors
- Split it into a factor pair (or divide by the smallest prime).
- Keep splitting until every factor is prime.
- Write the primes smallest first, in index form.
- Check by multiplying.
| \(40\) | \(=\) | \(2 \times 20\) |
| \(20\) | \(=\) | \(2 \times 10\) |
| \(10\) | \(=\) | \(2 \times 5\) |
\(40 = 2 \times 2 \times 2 \times 5 = 2^3 \times 5\). The prime factors are \(2\) and \(5\).
| \(2 \times 9 \times 7\) | \(=\) | \(126\) |
\(126 = 2 \times 3^2 \times 7\)
| \(200 \div 2\) | \(=\) | \(100\) |
| \(100 \div 2\) | \(=\) | \(50\) |
| \(50 \div 2\) | \(=\) | \(25\) |
| \(25 \div 5\) | \(=\) | \(5\) |
| \(5 \div 5\) | \(=\) | \(1\) |
\(200 = 2 \times 2 \times 2 \times 5 \times 5 = 2^3 \times 5^2\). Check: \(8 \times 25 = 200\).
| \(324\) | \(=\) | \(2 \times 2 \times 3 \times 3 \times 3 \times 3\) |
| \(\sqrt{324}\) | \(=\) | \(2 \times 3 \times 3 = 18\) |
\(324 = 2 \times 162 = 2 \times 2 \times 81\), and \(81 = 3 \times 3 \times 3 \times 3\). Split into two equal groups \(2 \times 3 \times 3\): \(\sqrt{324} = 18\).
(i) What is the number?
(ii) Is \(12\) a factor of \(N\)?
| \(2 \times 9 \times 25\) | \(=\) | \(450\) |
(i) \(450\)
(ii) \(12 = 2^2 \times 3\) needs two \(2\)s, but \(N\) has only one: no (\(450 \div 12 = 37.5\)).
(ii) What is the smallest whole number you can multiply \(150\) by to make a square number?
(iii) Find the square root of that square number.
| \(150 \times 6\) | \(=\) | \(900\) |
| \(2 \times 3 \times 5\) | \(=\) | \(30\) |
(i) \(150 = 2 \times 3 \times 5^2\)
(ii) \(2\) and \(3\) appear once, so multiply by \(2 \times 3 = 6\): \(900\)
(iii) \(\sqrt{900} = 2 \times 3 \times 5 = 30\)
Common pitfalls
Frequently asked questions
What is a product of prime factors?
A number written as primes multiplied together: \(90 = 2 \times 3^2 \times 5\).
What is a factor tree?
A diagram that splits a number into factor pairs until every branch ends in a prime.
Does it matter which factor pair you start with?
No. You always end with the same primes.
How do you find a square root with prime factors?
Split the primes into two equal groups and multiply one group.
Why is 1 not in a product of primes?
Because 1 is not a prime number.