Factors and Multiples: Why Children Swap Them, and How to Check

22 September 2026 · by Larry

Ask for the multiples of 6 and a child who knows her six times table perfectly well may write 1, 2, 3, 6. Ask for the factors of 6 and the same child may write 6, 12, 18, 24.

Nothing is wrong with her tables. Both lists come from exactly the same facts. What she is missing is which direction the question wants her to travel, and the two words give her no help at all, because "factor" and "multiple" sound equally official and neither one sounds like its meaning.

This is one of those topics where a child can be marked wrong on a question she could do in her sleep. The good news is that it is also one of the quickest to fix, once you know which of the slips below is happening.

The one picture that separates them

Take the fact 3 × 4 = 12. That single sentence says two things at once:

• 3 and 4 are factors of 12. They are the pieces that multiply together to make it.
• 12 is a multiple of 3, and a multiple of 4. It is what you land on when you count in threes, or in fours.

So factors go into a number, and multiples grow out of it. That gives a size rule a child can check without remembering any definition:

Factors are never bigger than the number. Multiples are never smaller.

It also gives the other difference that children find surprisingly helpful. The factors of 12 are a short, finished list: 1, 2, 3, 4, 6 and 12. That is all of them. The multiples of 12 are 12, 24, 36, 48 and on for ever. You can write out every factor; you can never write out every multiple.

Some families use "factors are few, multiples are many". Use whatever sticks, but tie it to the size rule, because the size rule is what she can actually check in an exam.

The check: "Is 24 a factor of 12, or a multiple of 12?" If she answers multiple and can say why ("it's bigger, it's 12 times 2"), she has the picture. If she guesses, start here before anything else.

Slip one: swapping the two words

This is the slip in the opening example. The child does the right kind of thinking in the wrong direction.

The fix is not more definitions. It is one question she asks herself before writing anything: should my answers be smaller or bigger than this number? Factors of 30 must all be 30 or smaller. Multiples of 30 must all be 30 or bigger. A list of multiples that contains a 5 has gone the wrong way, and she can see it without being told.

A second habit that helps: say the full sentence out loud. "3 is a factor of 12 because 3 times 4 is 12." "12 is a multiple of 3 because 3 times 4 is 12." Same fact, two sentences. Saying it both ways ties both words to one fact, and that link is exactly what was missing.

Slip two: missing factors

Ask for all the factors of 36 and many children will write something like 2, 3, 4, 6, 9 — and stop. Every number there is correct, and the answer still loses the mark, because four are missing.

Very often the two that go missing are the easiest ones: 1 and the number itself. Children do not think of them as "real" factors, because dividing by 1 or by the number feels like it does nothing. But 1 × 36 = 36, so both count, every time.

The others go missing because the child is hunting at random. The fix is to hunt in pairs, starting from 1 and working upward:

• 1 × 36
• 2 × 18
• 3 × 12
• 4 × 9
• 5 — does not go into 36, skip it
• 6 × 6 — the pair has met in the middle, so stop

Read the list down the left and back up the right and you have them all, in order: 1, 2, 3, 4, 6, 9, 12, 18, 36. Nine factors, none missed, and she knows she is finished because the pairs met. That last part matters. A child hunting at random never knows when to stop, so she either stops too early or keeps going and starts writing multiples.

The check: "Is 4 a factor of 30?" is really a division question, and it is the same idea as a division with a remainder. 30 ÷ 4 is 7 remainder 2. Anything left over means no. A factor divides in exactly, with nothing left.

Slip three: common factor or common multiple?

This is where the marks really go, because the words "factor" and "multiple" usually do not appear in the question at all. It is a story, and the child has to work out which one the story needs.

Two questions that look alike and are opposites:

A ribbon 24 cm long and another 36 cm long are cut into pieces of equal length, with nothing left over. What is the longest each piece can be?

One bus leaves the interchange every 12 minutes and another every 18 minutes. They both leave at 8 o'clock. When do they next leave together?

The ribbon question is about cutting something up into equal parts, so it needs a length that goes into both 24 and 36 exactly. That is a common factor, and the longest one is 12 cm.

The bus question is about things that repeat until they line up again. The first bus leaves at 12, 24, 36 minutes past; the second at 18, 36. They meet at 36 minutes, so the answer is 8.36. That is a common multiple.

Many assessment books (and many parents, from their own school days) call these the highest common factor and the lowest common multiple. Whatever words your child's school uses, the way to tell them apart is the same size rule from the start of this post:

• Cutting, sharing, splitting into equal groups, "the biggest possible" → the answer is no bigger than the smaller number. Factors.
• Repeating, "every so many minutes", "next time they meet", "the smallest number of" something that has to fit both → the answer is at least as big as the bigger number. Multiples.

If she gets 72 cm for the ribbon, or 6 minutes for the buses, the size rule catches it before the teacher does. The general habit of reading the story before choosing the sum is the same one covered in word problems when the sums are fine but the story is not.

Slip four: "it can't be both"

A smaller slip, but it unsettles children who otherwise have the topic. Is 12 a factor of 12, or a multiple of 12?

Both. 12 × 1 = 12, so 12 goes into itself exactly (a factor), and it is also the first number you land on counting in twelves (a multiple). Every number is a factor and a multiple of itself. And 1 is a factor of every whole number, which is why it belongs at the top of every factor list.

Children who have been told "factors are small, multiples are big" sometimes decide 12 has to be one or the other and change a correct answer. That is why the rule is worded as "never bigger" and "never smaller" rather than "smaller" and "bigger": the number itself sits on the line and belongs to both.

A two-minute check at the kitchen table

No worksheet. Just ask these, one at a time, and watch how she answers as much as what she answers:

• "Tell me three multiples of 7." Any three from 7, 14, 21, 28… is right. If she says 1 and 7, the words are swapped (slip one).
• "Tell me all the factors of 24." The answer is 1, 2, 3, 4, 6, 8, 12, 24. Watch whether she works in pairs or hunts at random, and whether 1 and 24 make it in (slip two).
• "Is 6 a factor of 40?" No: 40 ÷ 6 leaves 4 over. If she says yes because 40 is "in the six times table area", she is thinking about nearness rather than exactness.
• Read her the ribbon question above and ask only: factors or multiples? Do not let her calculate. If she can say "factors, because we're cutting it up", slip three is under control.

Often a child gets two or three of these right and stumbles on one. That one is the thing to work on. It is far more useful than another page of mixed questions, which mostly gives her more practice at the slip she already has.

What helps at home

Get the times tables solid first. Every factor question is a times table question read backwards. A child who has to count up to find 7 × 8 spends all her attention on the arithmetic and has none left for which direction she is meant to go. If the tables are shaky, that is the place to start, and factors will follow.

Make rectangles. Give her 12 small things — coins, Lego bricks, biscuits — and ask how many different rectangles she can arrange them into. 1 row of 12, 2 rows of 6, 3 rows of 4. The side lengths of those rectangles are exactly the factors of 12. Then try 7. She will find only one: a single long row. Some numbers just refuse to make a rectangle, and noticing that is real mathematics. It also connects directly to area, where the same rows-times-columns picture comes back.

Ask the size question, not the definition. When she hesitates, do not say "remember, a factor is…". Ask "should the answer be smaller or bigger than 30?" and let her work it out. A definition you give her is forgotten by Thursday; a check she can run herself goes into the exam with her.

What does not help

Drilling the two definitions until she can recite them. Plenty of children can recite both perfectly and still swap them under pressure, because a recited sentence is not the same thing as a picture. And do not rush to the word-problem version until the plain version is secure. If she is still swapping the words on "list the factors of 18", a bus timetable only adds a story on top of the confusion.

StudyLab's Primary 4 Factors & Multiples game asks one kind of question, over and over, on purpose: is this number a factor of that one? It is small and it is meant to be. That yes-or-no judgement, "does it go in exactly?", is the thing every harder factor question is built on.