Divisibility rules for 4, 6, 8, 9, 25 and 100 worksheet

Part A: Working out and filling in

1. 1. 8

   Add the three digits you know; □ must make the digit sum a multiple of 9.

   The digits shown add up to 4 + 6 + 0 = 10; adding 8 makes 18, a multiple of 9.

2. 2. 5

   Add the three digits you know; □ must make the digit sum a multiple of 9.

   The digits shown add up to 9 + 2 + 2 = 13; adding 5 makes 18, a multiple of 9.

3. 3. 2

   Check 4, 6, 8 and 9 in turn: look at the last two and last three digits of 2964, then its digit sum.

   Check each: 4, the last two digits; 8, the last three; 9, the digit sum; 6, even with a digit sum that is a multiple of 3. That makes 2.

4. 4. 6

   Only the last two digits matter: try □ from 9 downwards and see when they divide by 4.

   Only the last two digits matter: trying down from 9, 36 is the biggest that divides by 4, so □ is 6.

5. 5. 2

   Check 4, 6, 8 and 9 in turn: look at the last two and last three digits of 6060, then its digit sum.

   Check each: 4, the last two digits; 8, the last three; 9, the digit sum; 6, even with a digit sum that is a multiple of 3. That makes 2.

6. 6. 8

   Only the last two digits matter: try □ from 9 downwards and see when they divide by 4.

   Only the last two digits matter: trying down from 9, 88 is the biggest that divides by 4, so □ is 8.

Part B: Choose, or true or false

1. 7. False

   100 divides exactly by 4, so only the last two digits of 4385 matter.

   100 divides exactly by 4, so look only at the last two digits, 85: whether 85 ÷ 4 leaves anything tells you about 4385.

2. 8. A. 6825

   100 is a multiple of 25, so just look at each number's last two digits.

   100 is 4 times 25, so look at the last two digits: only 00, 25, 50 and 75 divide by 25.

3. 9. True

   100 divides exactly by 4, so only the last two digits of 7364 matter.

   100 divides exactly by 4, so look only at the last two digits, 64: whether 64 ÷ 4 leaves anything tells you about 7364.

4. 10. D. 3294

   Dividing by 6 means by both 2 and 3: check the number is even, then add its digits.

   Divisible by 6 means by both 2 and 3: 3294 is even and its digit sum is a multiple of 3.

5. 11. True

   1000 divides exactly by 8, so only the last three digits of 6704 matter.

   1000 divides exactly by 8, so look only at the last three digits, 704: whether 704 ÷ 8 leaves anything tells you about 6704.

6. 12. True

   1000 divides exactly by 8, so only the last three digits of 8680 matter.

   1000 divides exactly by 8, so look only at the last three digits, 680: whether 680 ÷ 8 leaves anything tells you about 8680.

Part C: Word problems

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

1. 13. 8 years

   Find the first year from 2006 on that divides by 4 (look at the last two digits), then count every 4th year.

   Use the last two digits to find the first and last years divisible by 4, then count every 4th year: 8.

2. 14. 1400 egg tarts

   To divide by both 4 and 25, what must the last two digits be? Then look between 1393 and 1454.

   Divisible by both 4 and 25, the last two digits must suit both, and only 00 does: between 1393 and 1454 that is 1400.

3. 15. 21 ml

   The whole hundreds fill cups exactly, so only the last two digits of 1521 matter.

   The whole hundreds fill cups exactly, so look at the last two digits, 21: dividing 21 by 25 leaves 21.

4. 16. 5 pens

   A number that divides by 6 is even and has a digit sum that is a multiple of three. Count on from 8989.

   Count on from 8989 to the first even number whose digit sum is a multiple of 3: 5 more pens.

5. 17. 8200 egg tarts

   To divide by both 4 and 25, what must the last two digits be? Then look between 8148 and 8217.

   Divisible by both 4 and 25, the last two digits must suit both, and only 00 does: between 8148 and 8217 that is 8200.

6. 18. 4 pens

   A number that divides by 6 is even and has a digit sum that is a multiple of three. Count on from 7454.

   Count on from 7454 to the first even number whose digit sum is a multiple of 3: 4 more pens.