Topics · Primary Maths · Factors & Multiples
Factors and multiples: the two directions of a times table
A factor divides into a number. A multiple is what you get by multiplying it. They are the same relationship read from opposite ends, and mixing them up is the whole difficulty.
- 265 exam papers analysed
- appears in 89% of P4 Maths papers
- ~3 min read
01 Same fact, two names
3 × 4 = 12 says two things at once. 3 and 4 are factors of 12; 12 is a multiple of 3 and of 4.
The way to keep them apart: factors are small and few — they fit inside the number and you can list them all. Multiples are big and endless — they run on past the number for ever.
A factor of 12 can never be more than 12. A multiple of 12 can never be less.
02 Listing factors without missing any
Work in pairs, starting from 1, and you cannot miss one. Each pair multiplies to the number.
- Start at 1. 1 × 12 = 12, so 1 and 12 are both factors.
- Try 2. 2 × 6 = 12 → 2 and 6.
- Try 3. 3 × 4 = 12 → 3 and 4.
- Try 4 — but 4 is already on the list, so the pairs have met and you are done. Factors of 12: 1, 2, 3, 4, 6, 12.
- Every number has 1 and itself as factors. Do not leave them out; papers count them.
03 Prime and composite
A prime number has exactly two factors: 1 and itself. 2, 3, 5, 7, 11, 13, 17, 19.
1 is not prime. It has only one factor — itself — and the definition asks for two. This is the single most-tested fact on the topic.
2 is prime, and it is the only even one, because every other even number has 2 as a factor.
A number with more than two factors is composite.
04 Common factors and common multiples
A common factor divides into both numbers. The highest common factor of 12 and 18 is 6, and it is what you divide by to simplify a fraction to its lowest terms.
A common multiple is in both times tables. The lowest common multiple of 4 and 6 is 12, and it is the common denominator you need to add quarters and sixths.
Those two jobs are why the topic matters: it is the machinery underneath fractions.
05 How to say them
These are words most children have never said out loud, and a word you cannot say is a word you avoid using. Capitals mark the syllable you lean on. Tap the speaker to hear it.
- factormore than onefactors
A whole number that divides exactly into another, leaving no remainder.
- multiplesayMUL-ti-puhlmore than onemultiples
The result of multiplying a number by a whole number. The multiples of 3 are 3, 6, 9, 12 and on for ever.
- prime numbermore than oneprime numbers
A number with exactly two factors, 1 and itself. 1 is not prime.
- composite numbersayKOM-puh-zitmore than onecomposite numbers
A number with more than two factors.
- highest common factor
The biggest number that divides into two numbers. Used to simplify fractions.
- lowest common multiple
The smallest number in both times tables. Used as a common denominator.
06 What each level is tested on
The same topic, asked differently as it goes up. The percentage is the share of papers at that level in which it appears.
- P2
Times tables and equal groups — the idea before the words for it.
Practise P2 Factors & Multiples — 10 questions →17% of P2 papers
- P3
Dividing exactly with no remainder, and skip counting through a times table.
Practise P3 Factors & Multiples — 10 questions →18% of P3 papers
- P4
Factors, multiples, and listing factor pairs.
Practise P4 Factors & Multiples — 10 questions →89% of P4 papers
- P5
Prime and composite numbers, common factors and common multiples.
Practise P5 Factors & Multiples — 10 questions →24% of P5 papers
- P6
Using highest common factor and lowest common multiple inside fraction work.
Practise P6 Factors & Multiples — 10 questions →45% of P6 papers
07 Mistakes worth knowing about
Each of these is wrong. Decide why before you open it.
“1 is a prime number.”
A prime has exactly two factors. 1 has only one, so it is neither prime nor composite. Papers ask this directly.
“The factors of 12 are 2, 3, 4 and 6.”
1 and 12 are factors too. Every number has 1 and itself, and leaving them out loses the mark.
“2 is not prime because it is even.”
2 is prime — its only factors are 1 and 2. It is the only even prime, which is what makes it worth remembering.
“The multiples of 3 are 1, 3 and 9.”
Those are factors of 9. Multiples of 3 are 3, 6, 9, 12, 15 and onwards without end.
“The highest common factor of 12 and 18 is 36.”
36 is the lowest common multiple. The highest common factor is 6 — a factor is never bigger than the numbers themselves.
Now try it
50 questions across 5 levels, marked as you go with the working shown.