Free to print, free to use

The divisibility rules poster

All eleven rules from 2 to 12, each with the quick test, a worked example and the mistake students actually make. It is grouped the way the rules are taught rather than in numerical order, so the pairs that get confused sit next to each other.

No sign-up, no email address, and nothing to give in exchange. Use it whether or not your department ever opens the game.

Read the rules in full

A3 for the wall, A4 for a desk. The sheet is the same on both, so choose the paper size in your printer dialogue and nothing else changes. On A3 the rule numbers read from about five metres, which covers a classroom; on A4 from about three and a half. It is designed to print in black and white, so a colour printer is not needed and nothing disappears without one.

Divisibility rules

Quick digit tests that tell you whether one number divides exactly by another, with no long division and no calculator.

The three tiers

Beginner2 · 3 · 5 · 10

Intermediate4 · 6 · 8 · 9 · 11 · 12

Advanced7

2 Last digit

The last digit is even: 0, 2, 4, 6 or 8.

3,978 ends in 8.
3,978 ÷ 2 = 1,989

5 Last digit

The last digit is 0 or 5.

2,335 ends in 5.
2,335 ÷ 5 = 467

10 Last digit

The last digit is 0.

4,290 ends in 0.
4,290 ÷ 10 = 429

3 Digit sum

Add the digits. If the digit sum is a multiple of 3, so is the number.

741 → 7 + 4 + 1 = 12, and 12 = 3 × 4.
741 ÷ 3 = 247

9 Digit sum

Add the digits. If the digit sum is a multiple of 9, so is the number.

6,453 → 6 + 4 + 5 + 3 = 18, and 18 = 9 × 2.
6,453 ÷ 9 = 717

Passing the 3 test is not enough. 24 has digit sum 6: a multiple of 3, not 9.

11 Plus, minus

From the last digit, add and subtract the digits in turn. 0 or a multiple of 11 means yes.

2,728 → 8 − 2 + 7 − 2 = 11.
2,728 ÷ 11 = 248

Not the plain digit sum (that is 3 and 9). −11 counts too.

4 Last two

Look at the last two digits. If they form a multiple of 4, so is the number.

1,916 → last two are 16, and 16 = 4 × 4.
1,916 ÷ 4 = 479

Two digits, not one. 34 ends in an even 4 and is not a multiple of 4.

8 Last three

Look at the last three digits. If they form a multiple of 8, so is the number.

5,120 → last three are 120, and 120 = 8 × 15.
5,120 ÷ 8 = 640

Three digits, not two. Passing the 4 test does not pass the 8 test.

6 Both tests

6 = 2 × 3: pass both tests. Even, and a digit sum that is a multiple of 3.

2 and 3 = 6

534 ends in 4, so it is even. 5 + 3 + 4 = 12, a multiple of 3. Both pass.
534 ÷ 6 = 89

One test alone is not enough. 15 passes 3 but is odd; 14 is even but fails 3.

12 Both tests

12 = 3 × 4: pass both tests. A digit sum that is a multiple of 3, and last two digits that make a multiple of 4.

3 and 4 = 12

348 → 3 + 4 + 8 = 15, a multiple of 3. Last two are 48, and 48 = 4 × 12.
348 ÷ 12 = 29

Not 2 and 6: they share a 2. 18 passes both and is not a multiple of 12.

7 Double and subtract Hardest

Drop the last digit, double it, subtract it from what is left. A multiple of 7 (including 0) means yes.

672 → drop the 2, double to 4.
67 − 4 = 63, and 63 = 7 × 9.
672 ÷ 7 = 96

3,101 → 310 − 2 = 308.
Again: 30 − 16 = 14 = 7 × 2.
3,101 ÷ 7 = 443

Repeat the step until you reach a number you know. For most three-digit numbers it is honestly quicker just to divide.

Divisible Dash

divisibledash.com/get

A maths game that drills these eleven rules until they are automatic, with free class boards for teachers. Worked examples for 2 to 12 at divisibledash.com/rules