Number
Grade 1-4Factors, Multiples & Primes
Factors, multiples and prime factorisation come up constantly, both as direct questions and as the hidden skill behind topics like HCF, LCM, simplifying fractions and surds. This lesson covers listing factors and multiples, finding common factors and multiples, and breaking a number down into its prime factors.
Written by Asad, Co-Founder of Teachably
Factors
A factor of a number divides into it exactly, with nothing left over. The factors of 18 are 1, 2, 3, 6, 9 and 18, because each of these divides into 18 with no remainder.
The quickest way to list all the factors of a number is to work through pairs: start at 1 and the number itself, then 2 and half of it (if it divides exactly), and keep working inwards until the pairs meet or cross over.
Multiples
A multiple of a number is what you get by multiplying it by a positive whole number. The first five multiples of 6 are 6, 12, 18, 24, 30, found by multiplying 6 by 1, 2, 3, 4 and 5.
Factors and multiples are often confused: a factor divides into a number, a multiple is what you get by multiplying it. 3 is a factor of 12, but 36 is a multiple of 12.
Common factors and common multiples
A common factor of two numbers divides into both of them exactly. 6 is a common factor of 12 and 18, because it divides into both.
A common multiple of two numbers appears in both of their times tables. 12 is a common multiple of 4 and 6, because both 4 and 6 divide into it.
Prime factorisation
Every whole number greater than 1 can be written as a product of prime numbers, and there is only ever one way to do it. This is usually found with a factor tree: split the number into any two factors, then keep splitting each factor until only prime numbers remain.
The final answer is written in index form, collecting repeated primes together. 60 splits into 2 × 30, then 2 × 2 × 15, then 2 × 2 × 3 × 5, which is written as 2² × 3 × 5.
Worked Examples
Three exam-style questions, fully solved.
Write down the first five multiples of 6.
Easy- 1.Multiply 6 by 1, 2, 3, 4 and 5 in turn
- 2.6 × 1 = 6, 6 × 2 = 12, 6 × 3 = 18, 6 × 4 = 24, 6 × 5 = 30
Answer: 6, 12, 18, 24, 30
Write down a common factor of 12 and 18, other than 1.
Medium- 1.List the factors of 12: 1, 2, 3, 4, 6, 12
- 2.List the factors of 18: 1, 2, 3, 6, 9, 18
- 3.Compare the two lists: 2, 3 and 6 all appear in both, so any one of them is a valid answer
Answer: 2, 3 or 6 (any one)
Express 72 as a product of its prime factors, giving your answer in index form.
Hard- 1.Split 72 into a pair of factors to start a factor tree: 72 = 2 × 36
- 2.Keep splitting each non-prime branch: 36 = 2 × 18, then 18 = 2 × 9, then 9 = 3 × 3
- 3.Collect every prime at the end of a branch: 72 = 2 × 2 × 2 × 3 × 3
- 4.Write repeated primes in index form
Answer: 2³ × 3²
Avoid These
The most common mistakes students make.
Confusing factors with multiples, for example listing the times table of a number when asked for its factors.
Forgetting to include 1 and the number itself when listing all the factors of a number.
Stopping a factor tree before every branch ends in a prime number, leaving a non-prime factor in the final answer.
Writing out the full multiplication instead of index form when index form is specifically asked for, for example 2 × 2 × 3 instead of 2² × 3.
When asked for "a common multiple," giving a multiple of only one of the two numbers instead of both.
FAQ
Questions parents and students ask.
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