Number System & Simplification • Topic 2 of 6

Divisibility Rules

Divisibility rules let you test a number without dividing. By 2: last digit even. By 3: digit sum divisible by 3. By 4: last two digits divisible by 4. By 5: ends in 0 or 5. By 6: divisible by 2 and 3. By 8: last three digits divisible by 8. By 9: digit sum divisible by 9. By 11: alternating digit-sum difference is 0 or a multiple of 11. For a missing digit, write the rule as an equation and solve.

✅ Solved examples

1. Is 738492 divisible by 6?
Last digit 2 -> divisible by 2. Digit sum 7+3+8+4+9+2 = 33 -> divisible by 3. So yes, divisible by 6.
2. For what digit x is 47x5 divisible by 9?
Digit sum 4+7+x+5 = 16+x must be a multiple of 9. 16+x = 18 -> x = 2.
3. Is 913040 divisible by 8?
Last three digits 040 = 40, and 40/8 = 5. Yes, divisible by 8.
4. Is 80593 divisible by 11?
Alternating sum from right: 3-9+5-0+8 = 7, not 0 or a multiple of 11, so not divisible by 11.

✏️ Practice — try these, take hints as needed

1. Find x if 12x4 is divisible by 4.
Only the last two digits x4 matter.
x4 divisible by 4 needs x even.
List 04,24,44,64,84.
x = 0,2,4,6 or 8
2. Is 54672 divisible by 8?
Last three digits 672.
672 / 8 = 84.
—
Yes
3. For what x is 3x52 divisible by 3?
Digit sum 3+x+5+2 = 10+x.
Need a multiple of 3.
10+x = 12, 15 or 18.
x = 2, 5 or 8
4. Is 94215 divisible by 11?
Alternating sum 5-1+2-4+9.
Compute.
Is it 0 or 11?
Yes (alternating sum = 11)
5. Largest 3-digit number divisible by both 8 and 12?
LCM(8,12) = 24.
Largest 3-digit multiple of 24.
24 x 41.
984

📝 Topic test — 8 questions

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