Divisibility Rules and Tests

Warning: Your screen size is quite small for viewing this. Rotate your device into landscape mode if you can.

1 Getting Started
2 Watch and Notice
3 Try It Together
4 Tick the Tests in Your Copy
5 Class Challenge
6 What Did We Notice?
7 What's Next
Pupil practice

1 - Getting Started ~4 mins

Illustration for Getting Started Here is a big number: 3,540. Without doing any dividing at all, can you already tell whether it splits evenly into 2, into 5, and into 10? Take a good look at it before anyone says a word. What is it about the number that gives the game away?

Teacher Notes

Give five seconds of quiet think-time before any hands go up, then take three hands-up answers, not open call-outs. Listen for pupils noticing the final 0 — that one digit is what settles all three. Don't confirm yet; the tests are built in the next step.

👇 Mark your steps as done as you complete them.

+5 XP

2 - Watch and Notice ~11 mins

Four hundred squares show the shading for one divisibility rule at a time. Look where the shading falls, and see which tests only need the last digit and which need something more.

Teacher Notes

Divisible by 2: even fills every second cell. Test is the last digit alone, 0, 2, 4, 6 or 8.

Divisible by 4: fewer cells shaded than for 2. Point at 14, even but not shaded, so even is not enough. Test uses last two digits. Work 36 (4 x 9) and 96 (4 x 24) on the grid, both shaded.

5 and 10 taught in prose, no grid. Ends in 5 or 0 divides by 5; ends in 0 divides by 10. Reason aloud: a number ending in 0 is a whole number of tens, and every ten divides by both, so only the last digit matters.

Divisible by 3: slanting stripe, so last digit is no help; shift to adding the digits. Work 24 (2 + 4 = 6) and 51 (5 + 1 = 6), both divide by 3. Light reason only: every ten, hundred and thousand is one more than a multiple of three. Don't chase a full proof.

Divisible by 9: steeper stripe, same digit-sum idea but total must divide by 9. Say 81, 8 + 1 = 9, slowly. Key distinction: digit-sum divides by 9 means divides by 9; digit-sum divides by 3 but not 9 means divides by 3 only.

Keep every worked example to a two-digit number they can find on the grid.

👇 Mark your steps as done as you complete them.

+10 XP

3 - Try It Together ~11 mins

Now we work through some divisors together on the grid: first we shade the multiples of 2, then 4, then 6. When we shade the multiples of two, every second cell lights up, because two goes into every even number. There will be fewer cells for four, and we will check each one with the last-two-digits test. Six is the interesting one. A multiple of six has to pass two tests at once, because 6 = 2 × 3: the number must be even (the 2 test) and its digit-sum must divide by 3 (the 3 test).

Key point

Before each pattern is revealed, put a hand up and say where you think it will land. One classmate does the shading at the board while everyone else predicts aloud. A wrong prediction is completely fine — this is thinking out loud, not a test.

Shade the multiples together

Teacher Notes

This round is for talking it through together — pupils take turns at the board and the class agrees or corrects out loud.

Multiples of 2 — a pupil shades; the class names the test (even last digit). Quick and confidence-building.
Multiples of 4 — pause here. Ask a pupil to justify one shaded number with the last-two-digits test (e.g. 36 → 36 divides by 4). Head off the common slip of thinking 'even means divides by 4' — point at 6 or 14, which are even but not shaded.
Multiples of 6 — this is the key one. Before revealing, ask the class to predict: 'which two tests does a number have to pass?' Draw out that 6 = 2 × 3. Point at 12 (passes both) versus 8 (even but digit-sum 8, fails the 3 test).

+15 XP

4 - Tick the Tests in Your Copy ~3 mins

COPYBOOK MOMENT

Before you open your copy, we will check one bigger number together for the 4 test so everyone sees how the last-two-digits rule works when the number has more than two digits.

Look at 144. Cover the hundreds digit with your finger; only the last two digits matter: 44. Does 44 divide by 4? Work it out together, then decide whether 144 itself divides by 4. That is the whole method: pull out the last two digits and test just those.

Now write these four numbers down the page in your maths copy, one under the other:

90
144
486
3,540

Beside each one, tick which of 2, 3, 4, 5, 9 and 10 it divides by. Next to every tick, write the test you used in a few words (for example, 'ends in 0' or 'digit-sum = 9' or 'last two digits divide by 4').

Teacher Notes

Board the worked example first: write 144, circle the last two digits 44, show 4 × 11 = 44, conclude it passes the 4 test. Then release the copybook list.

Walk the room glancing at whether pupils are naming the test, not just ticking — this is whole-class copybook practice, not marking. Watch for the 4 test being skipped or muddled with the 2 test; if a pupil ticks 4 for an even number without checking the last two digits, point them back to the board example.

+10 XP

5 - Class Challenge ~8 mins

Key point

Now we shade the numbers that pass each test in turn: first every multiple of 3, then every multiple of 6 (a number that passes both the 2 test and the 3 test), then every multiple of 9, and finally every multiple of 12. A multiple of 12 is a good one to finish on, because every one of them passes both the 3 test and the 4 test at the same time. Each round asks a little more of the digit-sum rule than the last.

Pass the test

Teacher Notes

These are the practice questions — pupils take turns at the board, check each answer, and the class confirms before moving on. Keep the board work brisk rather than over-explaining.

The ÷6 round is where the two-test idea gets its workout — confirm each shaded number is both even and has a digit-sum divisible by 3.
The ÷12 round rewards a pupil who separates the tests carefully. Revoice a good justification: 'the digit-sum divides by 3 and the last two digits divide by 4'. For example, 24 → 2 + 4 = 6 divides by 3, and 24 divides by 4.

+15 XP

Pupil practice

Module 2 · Operations and Computational Fluency Number

Lesson 24 · Divisibility Rules and Tests

Download Activity Book page (PDF)

123learn · Online learning platform

Unlock the full learning experience

You're previewing this lesson. Get full access to this lesson and hundreds more — each one ready to teach, with interactive activities, printable resources and pupil progress tracking built in.

Hundreds of curriculum-aligned lessons

Interactive activities in every lesson

Printable resources & progress tracking

Get started

Copyright Notice
This lesson is copyright of Coding Ireland 2017 - 2025. Unauthorised use, copying or distribution is not allowed.

Teacher Class Feed

1 - Getting Started ~4 mins

2 - Watch and Notice ~11 mins

3 - Try It Together ~11 mins

Shade the multiples together

4 - Tick the Tests in Your Copy ~3 mins

5 - Class Challenge ~8 mins

Pass the test

Unlock the full learning experience

XP