Numbers and operations · Ages 12–13

Divisibility by 4, 6, 8 and 9 worksheet

Part of: Working with numbers

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Numbers and operations · Ages 12–13

Numsprite
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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.Is 8271 divisible by 9? Its digit sum is , and the digit sum leaves when divided by 9.
Fill in each blank
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2.The four-digit number 361 is divisible by 6. What is the smallest digit that can go in ?
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3.The four-digit number 738 is divisible by 9. What is the smallest digit that can go in ?
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4.The four-digit number 223 is divisible by 9. What is the smallest digit that can go in ?
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5.The four-digit number 729 is divisible by 9. What digit goes in ?
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6.The four-digit number 496 is divisible by 9. What digit goes in ?
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13.A hall has 188 chairs to set out in rows of 8, with none left over. What is the fewest chairs that must be taken away?
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14.The five-digit number 8146 is divisible by both 4 and 9. What digit is ?
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15.4860 is a five-digit number divisible by 6. What is the largest digit that can go in ?
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16.A tour group of 56 people travels in minibuses that seat 6 each. If every minibus must be full, at least how many more people must join?
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17.Tsz-hin has 2166 sweets to share equally among 6 friends with none left over. Check it: 2166 ÷ 2 leaves , the digit sum of 2166 is , and the digit sum ÷ 3 leaves .
Fill in each blank
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18.A number is divisible by 4, 6 and 9, and lies between 971 and 1005. What is the number?
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Show the answers and working

Divisibility by 4, 6, 8 and 9 worksheet: Answers and working

1. 18, 0
8 + 2 + 7 + 1 = 18, and 18 ÷ 9 leaves 0. Only a remainder of 0 means divisible by 9.
2. 2
Divisible by 6 means divisible by both 2 and 3: □ must be even, and 10 + □ a multiple of 3. The least is 2.
3. 0
The known digits add up to 7 + 3 + 8 = 18. With □ added the sum must be a multiple of 9, so the least is 0.
4. 2
The known digits add up to 2 + 2 + 3 = 7. With □ added the sum must be a multiple of 9, so the least is 2.
5. 9
The known digits add up to 7 + 2 + 9 = 18. With □ the sum must be a multiple of 9, so it is 27 and □ is 9. A four-digit number can't start with 0.
6. 8
The known digits add up to 4 + 9 + 6 = 19. With □ the sum must be a multiple of 9, so it is 27 and □ is 8. A four-digit number can't start with 0.

7. True
Only the last two digits matter: 64 ÷ 4 leaves 0. The whole number is divisible by 4 only when they make a multiple of 4 (remainder 0).
8. False
The digits add up to 8 + 9 + 9 + 4 = 30, and 30 ÷ 9 leaves 3. A number is divisible by 9 only when its digit sum is a multiple of 9.
9. True
The digits add up to 2 + 4 + 1 + 2 = 9, and 9 ÷ 9 leaves 0. A number is divisible by 9 only when its digit sum is a multiple of 9.
10. D. 3344
Its last three digits, 344 = 8 × 43, make a multiple of 8, so 3344 divides by 8.
11. A. 2022
2022 is even and its digit sum is a multiple of 3: divisible by both 2 and 3, so by 6.
12. A. 1989
1989's digit sum is a multiple of 9, so 9 divides it; but it is odd, so 2 doesn't, and nor does 6.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 4 chairs
188 ÷ 8 leaves 4: take 4 away and the rest fill the rows exactly.
14. 8
For 9, 19 + □ must be a multiple of 9; for 4, the last two digits 6□ must make a multiple of 4. Only 8 does both.
15. 6
For 6, □ must be even and 18 + □ a multiple of 3. Trying even digits down from 8, the largest is 6.
16. 4 people
56 ÷ 6 leaves 2, so 6 − 2 = 4 more fill one more minibus.
17. 0, 15, 0
Divisible by 6 means divisible by 2 (remainder 0) and by 3 (digit sum ÷ 3 leaves 0). The digits of 2166 add up to 15.
18. 972
Divisible by 4, 6 and 9 means a multiple of 36, their L.C.M.; between 971 and 1005 the only one is 972.

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