Mental Math · Level Up · L5

Divisible by 7 or 11?

Seven and eleven have their own quick tests. For 11, add the digits with alternating + and − signs; if that's a multiple of 11, so is the number. For 7, repeatedly drop the last digit, double it, and subtract — until you reach something you recognise.

01 / THE METHOD

How to think about it.

Watch the method

Test for 7 and 11

308: alternating digit sum 8 − 0 + 3 = 11, a multiple of 11 → divisible by 11. And the 7-test confirms 308 = 7 × 44, so it's divisible by 7 too.

1 / 4

308

= ?

+3-0+8 = 11

For 11: add the digits with alternating signs (+ − + −).

11 \to 11?

11 is a multiple of 11 → divisible by 11.

308 \to 7?

For 7: drop the last digit, double it, subtract from the rest; repeat until small.

308 \div 7 = 44

Checking, 308 is divisible by 7.

— / WHY IT WORKS

The idea underneath.

Alternating sum (11) and double-subtract (7)

For 11: +,−,+,− across the digits; a result of 0 or ±11 means yes. For 7: take off the last digit, double it, subtract it from the remaining number; repeat. Reaching 0 or a known multiple of 7 means yes.

Why it works

Both rules shrink a big number to a tiny one that keeps the same divisibility.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Is 253 divisible by 11?

Alternating

3 − 5 + 2 = 0.

Check

0 is a multiple of 11 → yes (253 = 11 × 23).

EX 2medium

Is 91 divisible by 7?

Double-subtract

9 − (2×1) = 7.

Check

7 is divisible by 7 → yes (91 = 7 × 13).

EX 3hard

Is 2728 divisible by 11?

Alternating

8 − 2 + 7 − 2 = 11.

Check

11 is a multiple of 11 → yes.

04 / COMMON TRAPS

Where students slip.

Trap

Using the digit-sum (÷3, ÷9) rule for 7 or 11.

Fix

Seven and eleven need their own tests. The plain digit sum tells you nothing about them.

Trap

Adding all digits for 11 instead of alternating.

Fix

For 11 the signs alternate: + − + −. A plain sum is the wrong test.

05 / QUIZ

Test yourself.

The Test

12 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.

0/12Answered

How to answerType just the value. Formatting won't trip you up — we ignore spaces, capital letters, commas, and subscript style, so t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).

Core

Core (4)

Is 308 divisible by 11? Type 1 for yes, 0 for no.

Need a hint?

For 11, add the digits with alternating + and − signs.
For 7, double the last digit and subtract it from the rest.

Is 253 divisible by 11? Type 1 for yes, 0 for no.

Need a hint?

For 11, add the digits with alternating + and − signs.
For 7, double the last digit and subtract it from the rest.

Is 341 divisible by 11? Type 1 for yes, 0 for no.

Need a hint?

For 11, add the digits with alternating + and − signs.
For 7, double the last digit and subtract it from the rest.

Is 482 divisible by 11? Type 1 for yes, 0 for no.

Need a hint?

For 11, add the digits with alternating + and − signs.
For 7, double the last digit and subtract it from the rest.

Apply

Apply (4)

Is 84 divisible by 7? Type 1 for yes, 0 for no.

Need a hint?

For 11, add the digits with alternating + and − signs.
For 7, double the last digit and subtract it from the rest.

Is 91 divisible by 7? Type 1 for yes, 0 for no.

Need a hint?

For 11, add the digits with alternating + and − signs.
For 7, double the last digit and subtract it from the rest.

Is 95 divisible by 7? Type 1 for yes, 0 for no.

Need a hint?

For 11, add the digits with alternating + and − signs.
For 7, double the last digit and subtract it from the rest.

Is 119 divisible by 7? Type 1 for yes, 0 for no.

Need a hint?

For 11, add the digits with alternating + and − signs.
For 7, double the last digit and subtract it from the rest.

Extension

Extension (4)

Is 1234 divisible by 11? Type 1 for yes, 0 for no.

Need a hint?

For 11, add the digits with alternating + and − signs.
For 7, double the last digit and subtract it from the rest.

Is 2728 divisible by 11? Type 1 for yes, 0 for no.

Need a hint?

For 11, add the digits with alternating + and − signs.
For 7, double the last digit and subtract it from the rest.

Is 343 divisible by 7? Type 1 for yes, 0 for no.

Need a hint?

For 11, add the digits with alternating + and − signs.
For 7, double the last digit and subtract it from the rest.

Is 539 divisible by 11? Type 1 for yes, 0 for no.

Need a hint?

For 11, add the digits with alternating + and − signs.
For 7, double the last digit and subtract it from the rest.