Factors and Multiples • Topic 6 of 7

Divisibility Rules

Divisibility rules let you test factors without long division. A number is divisible by 2 if it ends in an even digit; by 3 if its digit sum is divisible by 3; by 4 if its last two digits form a multiple of 4; by 5 if it ends in 0 or 5; by 6 if it passes both the 2 and 3 tests; by 8 if its last three digits are a multiple of 8; by 9 if its digit sum is divisible by 9; by 10 if it ends in 0; and by 11 if the alternating digit sum is a multiple of 11. Combining rules (for 6, 12, 15, 18) is a common SAT move.

✅ Solved examples

1. Is 4,572 divisible by 6?
Test 2 and 3. It ends in 2 (even) ✓. Digit sum 4+5+7+2 = 18, divisible by 3 ✓. Passing both, 4,572 is divisible by 6.
2. Is 8,135 divisible by 11?
Alternating sum from the right: 5 − 3 + 1 − 8 = −5. Since −5 is not a multiple of 11, 8,135 is NOT divisible by 11.
3. What digit d makes 23d divisible by 9?
Digit sum 2 + 3 + d = 5 + d must be a multiple of 9. The value 9 gives d = 4. So d = 4 (the number 234).
4. Is 7,128 divisible by 8?
Check the last three digits, 128. Since 128 = 8 × 16, it is a multiple of 8, so 7,128 is divisible by 8.

✏️ Practice — try these, take hints as needed

1. Is 3,456 divisible by 4?
Only the last two digits matter for 4.
Look at 56.
56 ÷ 4 = 14.
Yes.
2. Is 1,234 divisible by 3?
Add the digits.
1+2+3+4 = 10.
10 is not a multiple of 3.
No.
3. What is the largest digit d so that 5,d2 (i.e. 5d2) is divisible by 6? Hint: it must be even already.
It ends in 2, so the 2-test passes.
Need digit sum 5 + d + 2 = 7 + d divisible by 3.
Largest single digit d making 7+d a multiple of 3 is d = 8 (sum 15).
d = 8.
4. Is 90,909 divisible by 11?
Take the alternating sum of digits.
9 − 0 + 9 − 0 + 9 = 27.
Is 27 a multiple of 11?
No (27 is not a multiple of 11).
5. A number is divisible by both 5 and 9. What is the smallest 3-digit such number?
Divisible by 5 and 9 means divisible by 45.
Find the smallest 3-digit multiple of 45.
45 × 3 = 135.
135.

📝 Topic test — 8 questions

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