27 July 2016

Divisibility Rules

In the last lesson of term my students played a few rounds of a Countdown type game, and were particularly stumped by this problem:
I managed to do it pretty quickly and briefly experienced one of those lovely moments of appearing, to my students at least, to be a maths genius. Of course all I did was spot that 531 divides by 9, then it was straightforward. Have a go.

A colleague asked me how I'd done it so quickly and I told her that I'd used divisibility rules. She said that she'd never taught divisibility rules because she'd never seen it specified on a scheme of work. It strikes me that this is a helpful bit of mathematical knowledge that many secondary maths teachers don't teach. Do you teach it? In what year? Most resources for this topic are aimed at primary children but I think we should probably revisit it at Key Stage 3.

I was aware that my Year 10s didn't know the divisibility rules, so I covered them as part of the 'Factors and Multiples' topic this year (ie alongside prime factorisation, highest common factor etc). It's a good way to review the fundamentals of multiplication and to develop fluency and efficiency with numbers. To my Year 10s the rules seemed like 'new' maths that they'd not seen before (or if they had, they couldn't recall it), so it made an interesting and suitably challenging lesson.

I also ran a session on divisibility rules with some smart Year 3s and 4s at a local primary school this year. They picked it up well, and again I saw it as a good way to develop their understanding of multiplication and their number fluency.

So it's a topic that works well with any age group, from Key Stage 2 to Key Stage 4. Let's take a quick look at the rules and resources.

The Rules
Most children will easily be able to determine whether a number is divisible by 2, 5 or 10. The neat 'tricks' are for 3 (the digit sum is divisible by 3) and 9 (the digit sum is divisible by 9). Once we know whether a number divides by 3, we know whether it divides by 6 (ie all even multiples of 3 are multiples of 6). For divisibility by 4 there are two alternatives: either check whether the last two digits divide by 4, or halve the number and see if your answer is even (the four times table being double the two times table). The seven key tests are shown in the graphic below (there are loads of nice graphics for this on google images).

I didn't bother teaching the rule for divisibility by 7 because it's not straightforward. Rather then memorise this rule I thought my students would be better off just checking for divisibility by 7 with long division.

If you're interested in all the rules, from 1 - 30 and beyond, check out the Wikipedia page Divisibility rule.

I found a mixture of uninspiring worksheets and bizarre activities when I searched for resources online (the more unusual activities included Divisibility Rock n' RuleNFL divisibility dance and I'll take, you take...).

For the interactive whiteboard we have Vectorkids: divisibility rules, Divisibility Test and Delightfully Divisible.

If you're looking for a well structured worksheet pack, this is quite good.

This simple PowerPoint sets out the rules and contains practice activities mainly drawn from this homeschool website. It's nothing special but feel free to borrow and adapt it. It didn't take a whole lesson so it's worth adding some more challenging problems, such as this task from Don Steward.
Don Steward also has slides on divisibility rules, full of lovely challenging problems.

Dan Walker has a great set of slides and activities on divisibility rules too, including this lovely Venn activity.
Why do the rules work?
The ancient Greeks knew rules for divisibility by 2, 3, 5 and 9 in the third century BC.

So why does the digit sum of multiples of three divide by three? Sal Khan explains here...


He has a similar video for divisibility by 9.

Do let me know about your experiences of teaching divisibility rules and any resource recommendations. If you've not taught it before, have a go next year. It's useful knowledge and well worth teaching.


  1. Thanks to Kate Brewer who emailed to recommend this interactive Pascal's Triangle. Read this post for more detail.

  2. What a wonderful resource! Thank you! Just as a heads up the "simple divisibility game" is full of incorrect answers with the 9's divisibility rule. Double check me... http://www.oswego.org/ocsd-web/match/dragflip.asp?filename=slanedivrules

    1. Thank you - you're right! I'll remove it from the post. Cheers.

  3. https://www.youtube.com/watch?v=UMIt8eVJ0sI

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