Divisibility Tests

Technique: Divisibility shortcuts

For each question, decide whether the number is divisible by the given divisor. Type 1 for YES, 0 for NO. The divisor cycles randomly across six rules — pick the right shortcut for each.

The six rules

• ÷3 — digit sum is divisible by 3. 4731 → 4+7+3+1=15 → yes (15 = 3×5).
• ÷4 — last two digits divisible by 4. 1232 → 32 → yes (32 = 4×8). Ignore everything else.
• ÷7 — subtract 2×(last digit) from the rest, repeat until small. 4732 → 473 − 2×2 = 469 → 46 − 2×9 = 28 → yes (28 = 7×4).
• ÷8 — last three digits divisible by 8. 4072 → 072 = 72 → yes (72 = 8×9).
• ÷9 — digit sum divisible by 9. 4329 → 4+3+2+9=18 → yes (18 = 9×2).
• ÷11 — alternating digit sum divisible by 11. 4719 → 4−7+1−9 = −11 → yes.

Why these shortcuts work

Each rule comes from how the divisor relates to powers of 10. 10 ≡ 1 (mod 9), so each digit contributes its face value to n mod 9 — that’s the digit-sum rule for 3 and 9. 100 ≡ 0 (mod 4), so only the last two digits matter for ÷4. 10 ≡ −1 (mod 11), so each successive digit flips sign — alternating sum.

The ÷7 trick uses 10n + d ≡ 0 (mod 7) ⇔ n − 2d ≡ 0 (mod 7) — repeatedly chops the trailing digit until you reach a number you can recognize.

Every question has a 30 second shot clock.