Tests of Divisibility

6, 8, 9, 10, and 11

In this lesson:

• Apply five divisibility tests without long division
• Recognise when a number passes or fails each test
• Use multiple tests together on the same number

What You Will Learn This Lesson

Five tests — no long division:

1. 6-test: divisible by both 2 and 3
2. 8-test: last three digits divisible by 8
3. 9-test: digit sum divisible by 9
4. 10-test: last digit is 0
5. 11-test: alternating digit sum = 0 or divisible by 11

Can Nalongo Make Groups of Six?

Nalongo is a market trader. She has 534 matooke bananas.

Can she make equal bunches of 6?

What tests do you already know?

• 2-test: divisible by 2 if even
• 3-test: divisible by 3 if digit sum divisible by 3

Test 534: Does It Pass Both Checks?

2-test: Last digit 4 → even → Yes

3-test: 5 + 3 + 4 = 12 → 12 ÷ 3 = 4 → Yes

534 passes both tests.

What does passing both tests tell you about divisibility by 6?

The 6-Test: Why It Works

Divisible by 6 = divisible by both 2 AND 3.

6 = 2 × 3 and 2, 3 share no common factors:

• Divisible by 2 → factor of 2
• Divisible by 3 → factor of 3
• Both → factor of 6

Applying the 6-Test: Three Worked Examples

534: Even, digit sum 12 ÷ 3 = 4 → Divisible by 6

342: Even, digit sum 9 ÷ 3 = 3 → Divisible by 6

615: Odd → fails 2-test → Not divisible by 6

Both tests must pass. One green tick is not enough.

The 9-Test and the 10-Test

9-test: digit sum divisible by 9 (stricter than the 3-test).

• 729: 7+2+9 = 18 ÷ 9 = 2 → Divisible by 9
• 5,049: 5+0+4+9 = 18 ÷ 9 = 2 → Divisible by 9

10-test: last digit is 0.

Quick Check: 9-Test and 6-Test

Is 3,924 divisible by 9?

Add the digits: 3 + 9 + 2 + 4 = ?

Work it out — then decide: divisible by 9 or not?

Bonus: Is 3,924 also divisible by 6? What two checks do you need?

Commit to your answers before the next slide.

The 8-Test and 11-Test Work Differently

Tests for 3, 9 use digit sums; tests for 2, 5, 10 use the last digit.

For 8 and 11, the method changes:

• 8-test: last three digits (not digit sum)
• 11-test: alternating + and − digit sums

Warning: do NOT apply digit-sum thinking to the 8-test.

The 8-Test: Make Your Prediction First

Before I explain the 8-test, here is a number: 126.

Someone says: "126 has digit sum 9 — so it must be divisible by 8."

Do you agree? Predict: is 126 ÷ 8 an exact division?

Think before the next slide.

Why the 8-Test Uses Last Three Digits

Key fact: 1000 = 8 × 125

1,000 is exactly divisible by 8 — so any thousands portion is too.

Only the last three digits determine divisibility by 8.

126 ÷ 8 = 15.75 — NOT divisible by 8, despite digit sum 9!

Applying the 8-Test: Two Worked Examples

5,472: last three digits 472 → 472 ÷ 8 = 59 (exact) → Divisible by 8

3,918: last three digits 918 → 918 ÷ 8 = 114.75 (not exact) → Not divisible by 8

Isolate the last three digits, divide by 8, check if exact.

The 11-Test: Write Signs First

Divisible by 11: alternating digit sum (left to right: +, −, +, −, …) = 0 or divisible by 11.

Write the signs below the digits before computing:

Applying the 11-Test: Three Worked Examples

946: +9 − 4 + 6 = 11 → Divisible by 11

84,953: +8 − 4 + 9 − 5 + 3 = 11 → Divisible by 11

13,475: +1 − 3 + 4 − 7 + 5 = 0 → Divisible by 11

Result 0 = divisible by 11.

Quick Check: Apply the 11-Test Now

Apply the 11-test to 5,368:

Write the signs: + 5 − 3 + 6 − 8 = ?

Compute and decide: divisible by 11 or not?

Then check: is 5,368 also divisible by 8? (What are the last three digits?)

Common Errors: Two to Watch For

8-test error: 126 digit sum 9 → divisible by 8?
No — 126 ÷ 8 = 15.75. Digit-sum works for 3, 9 only.

11-test error: 9 + 4 + 6 = 19 → not divisible by 11?
No — 9 − 4 + 6 = 11 — divisible by 11.

Applying All Five Tests Together

Start with 990 — next slide.

All Five Tests Applied to 990

• ÷6: even, digit sum 18 ÷ 3 = 6 → Yes
• ÷8: 990 ÷ 8 = 123.75 → No
• ÷9: 18 ÷ 9 = 2 → Yes
• ÷10: ends in 0 → Yes
• ÷11: 9−9+0 = 0 → Yes

Your Turn: Complete the Table

In pairs, complete the table for 8,712 and 35,640.

Apply all five tests. Record Y or N.

When done, verify two cells by doing the actual division.

No scaffolding from here — use the tests you know.

Five Tests: What to Remember

• 6-test: divisible by both 2 AND 3
• 9-test: digit sum divisible by 9
• 10-test: last digit is 0
• 8-test: last three digits divisible by 8
• 11-test: alternating digit sum = 0 or divisible by 11

Watch out: digit-sum fails for 8; 11-test alternates signs.