Year 5: Tests of divisibility
Part of the Year 5 prime numbers, factors, squares and cubes chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Tests of divisibility
A number is divisible by another number if you can divide it exactly, with nothing left over. 24 is divisible by 6 because 24 ÷ 6 = 4. This is the same as saying that 24 is a multiple of 6. But what about a big number like 3,276? Is it divisible by 4? You do not need to divide. A test of divisibility gives a quick answer by looking at the digits.
| A number is divisible by… | if… | Example |
|---|---|---|
| 2 | The last digit is 0, 2, 4, 6 or 8 (an even number). | 3,846 ends in 6 → yes |
| 3 | The digits add up to a number in the 3 times table. | 4,263: 4+2+6+3 = 15 → yes |
| 4 | The last two digits make a number in the 4 times table. (Or halve twice.) | 2,716: 16 is in the 4s → yes |
| 5 | The last digit is 0 or 5. | 1,235 ends in 5 → yes |
| 6 | It is divisible by 2 and by 3. | 438 is even; 4+3+8 = 15 → yes |
| 8 | Halve it three times and you always get a whole number. (Or the last three digits are in the 8s.) | 648 → 324 → 162 → 81 → yes |
| 9 | The digits add up to a number in the 9 times table. | 7,893: 7+8+9+3 = 27 → yes |
| 10 | The last digit is 0. | 5,430 ends in 0 → yes |
| 25 | The last two digits are 00, 25, 50 or 75. | 1,375 ends in 75 → yes |
| 100 | The last two digits are 00. | 4,500 ends in 00 → yes |
Why they work. Every ten is even, and every ten is in the 5 times table, so only the last digit matters for 2, 5 and 10. Every hundred is in the 4 times table and the 25 times table, so only the last two digits matter for 4, 25 and 100. For 3 and 9 the sum of the digits tells you. The tests for 6 and 8 use the ones you already know: 6 = 2 × 3, and 8 means halving three times.
Key ideas
- 2, 5, 10: look at the last digit.
- 4, 25, 100: look at the last two digits (for 4: halve twice).
- 8: halve three times and stay whole.
- 3 and 9: add up all the digits. Is the total in the 3 (or 9) times table?
- 6: divisible by 2 and by 3. (You can combine two tests when the numbers share no factor except 1: for 12 use 3 and 4.)
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for each number in turn. Write down which tests pass.
- Choose the tests. Each number has its own quick test.2, 5, 10
- Do each test on 4,386. Look only at the digits the test needs.
Test Working Divisible? 2 The last digit of 4,386 is 6. Is it 0, 2, 4, 6 or 8? yes. yes 5 The last digit of 4,386 is 6. Is it 0 or 5? no. no 10 The last digit of 4,386 is 6. Is it 0? no. no - List the numbers that pass. 2
Check. Divide one to check: 4,386 ÷ 2 = 2,193. ✓
Watch out. Do every test. A number can pass some tests and fail others.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 3. It saves doing the whole division.
- Remember the test for 3. Every 10, 100, 1000 is one more than a multiple of 3 (9, 99, 999), so only the digit total matters.Add up the digits. If the total is in the 3 times table, the number is divisible by 3.
- Use the test on 738. Only look at the digits that the test needs.7 + 3 + 8 = 18. 18 is in the 3 times table (3 × 6 = 18). yes.
- Answer the question. 738 is divisible by 3: Yes
Check. Divide to check: 738 ÷ 3 = 246. No remainder. ✓
Watch out. Add all the digits, not just the last one.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 4. It saves doing the whole division.
- Remember the test for 4. Every 100 is in the 4 times table, so only the last two digits matter.Look at the last two digits. If they make a number in the 4 times table, the number is divisible by 4. (Or halve it twice.)
- Use the test on 516. Only look at the digits that the test needs.The last two digits are 16. 16 ÷ 4 = 4. yes.
- Answer the question. 516 is divisible by 4: Yes
Check. Divide to check: 516 ÷ 4 = 129. No remainder. ✓
Watch out. Use the last two digits, not just the last one: 2,136 is divisible by 4 because 36 is.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 8. It saves doing the whole division.
- Remember the test for 8. 8 is 2 × 2 × 2, so we can halve three times.Halve the number three times. If you get a whole number every time, it is divisible by 8.
- Use the test on 7,240. Only look at the digits that the test needs.7,240 ÷ 2 = 3,620 , 3,620 ÷ 2 = 1,810 , 1,810 ÷ 2 = 905. yes.
- Answer the question. 7,240 is divisible by 8: Yes
Check. Divide to check: 7,240 ÷ 8 = 905. No remainder. ✓
Watch out. Halve three times. If you can only halve twice, the number is a multiple of 4 but not of 8.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 6. It saves doing the whole division.
- Remember the test for 6. 6 is 2 × 3, so the number must pass both tests.The number is divisible by 2 and by 3.
- Use the test on 534. Only look at the digits that the test needs.Test 2: The last digit of 534 is 4. Is it 0, 2, 4, 6 or 8? yes.
Test 3: 5 + 3 + 4 = 12. 12 is in the 3 times table (3 × 4 = 12). yes.
Both tests must pass: yes. - Answer the question. 534 is divisible by 6: Yes
Check. Divide to check: 534 ÷ 6 = 89. No remainder. ✓
Watch out. Both tests must pass. A number that is only divisible by 2, or only by 3, is not divisible by 6.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 9. It saves doing the whole division.
- Remember the test for 9. Every 10, 100, 1000 is one more than a multiple of 9 (9, 99, 999), so only the digit total matters.Add up the digits. If the total is in the 9 times table, the number is divisible by 9.
- Use the test on 6,372. Only look at the digits that the test needs.6 + 3 + 7 + 2 = 18. 18 is in the 9 times table (9 × 2 = 18). yes.
- Answer the question. 6,372 is divisible by 9: Yes
Check. Divide to check: 6,372 ÷ 9 = 708. No remainder. ✓
Watch out. Add all the digits, not just the last one.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for each number in turn. Write down which tests pass.
- Choose the tests. Each number has its own quick test.25, 100
- Do each test on 1,375. Look only at the digits the test needs.
Test Working Divisible? 25 The last two digits are 75. Is it 00, 25, 50 or 75? yes. yes 100 The last two digits are 75. Is it 00? no. no - List the numbers that pass. 25
Check. Divide one to check: 1,375 ÷ 25 = 55. ✓
Watch out. Do every test. A number can pass some tests and fail others.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. 12 is 3 × 4, and 3 and 4 have no common factor except 1. So a number is divisible by 12 if it passes both the 3 test and the 4 test.
- Use the test for 3. Add up the digits. If the total is in the 3 times table, the number is divisible by 3.4 + 5 + 7 + 2 = 18. 18 is in the 3 times table (3 × 6 = 18). yes.
- Use the test for 4. Look at the last two digits. If they make a number in the 4 times table, the number is divisible by 4. (Or halve it twice.)The last two digits are 72. 72 ÷ 4 = 18. yes.
- Put the two tests together. It must pass both tests.yes and yes: 4,572 is divisible by 12.
Check. Divide: 4,572 ÷ 12 = 381. ✓
Watch out. This only works when the two numbers share no factor except 1. Do not use 2 and 4 to test for 8.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for each number in turn. Write down which tests pass.
- Choose the tests. Each number has its own quick test.2, 3, 4, 5, 6, 8, 9, 10
- Do each test on 360. Look only at the digits the test needs.
Test Working Divisible? 2 The last digit of 360 is 0. Is it 0, 2, 4, 6 or 8? yes. yes 3 3 + 6 + 0 = 9. 9 is in the 3 times table (3 × 3 = 9). yes. yes 4 The last two digits are 60. 60 ÷ 4 = 15. yes. yes 5 The last digit of 360 is 0. Is it 0 or 5? yes. yes 6 Test 2: The last digit of 360 is 0. Is it 0, 2, 4, 6 or 8? yes.
Test 3: 3 + 6 + 0 = 9. 9 is in the 3 times table (3 × 3 = 9). yes.
Both tests must pass: yes.yes 8 360 ÷ 2 = 180 , 180 ÷ 2 = 90 , 90 ÷ 2 = 45. yes. yes 9 3 + 6 + 0 = 9. 9 is in the 9 times table (9 × 1 = 9). yes. yes 10 The last digit of 360 is 0. Is it 0? yes. yes - List the numbers that pass. 2, 3, 4, 5, 6, 8, 9, 10
Check. Divide one to check: 360 ÷ 2 = 180. ✓
Watch out. Do every test. A number can pass some tests and fail others.
Watch out
For 4 you need the last two digits, not just the last one: 2,136 is divisible by 4 because 36 is. For 3 and 9 add all the digits. A number that ends in 5 is divisible by 5, but only some of them (the ones ending in 25 or 75, and 00, 50) are divisible by 25. And do not test for 8 by checking for 2 and 4: 12 passes both but is not divisible by 8.Exercise 3B Tests of divisibility
Non-calculatorKey words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for each number in turn. Write down which tests pass.
- Choose the tests. Each number has its own quick test.2, 5, 10
- Do each test on 1,230. Look only at the digits the test needs.
Test Working Divisible? 2 The last digit of 1,230 is 0. Is it 0, 2, 4, 6 or 8? yes. yes 5 The last digit of 1,230 is 0. Is it 0 or 5? yes. yes 10 The last digit of 1,230 is 0. Is it 0? yes. yes - List the numbers that pass. 2, 5, 10
Check. Divide one to check: 1,230 ÷ 2 = 615. ✓
Watch out. Do every test. A number can pass some tests and fail others.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for each number in turn. Write down which tests pass.
- Choose the tests. Each number has its own quick test.2, 5, 10
- Do each test on 4,375. Look only at the digits the test needs.
Test Working Divisible? 2 The last digit of 4,375 is 5. Is it 0, 2, 4, 6 or 8? no. no 5 The last digit of 4,375 is 5. Is it 0 or 5? yes. yes 10 The last digit of 4,375 is 5. Is it 0? no. no - List the numbers that pass. 5
Check. Divide one to check: 4,375 ÷ 5 = 875. ✓
Watch out. Do every test. A number can pass some tests and fail others.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for each number in turn. Write down which tests pass.
- Choose the tests. Each number has its own quick test.2, 5, 10
- Do each test on 508. Look only at the digits the test needs.
Test Working Divisible? 2 The last digit of 508 is 8. Is it 0, 2, 4, 6 or 8? yes. yes 5 The last digit of 508 is 8. Is it 0 or 5? no. no 10 The last digit of 508 is 8. Is it 0? no. no - List the numbers that pass. 2
Check. Divide one to check: 508 ÷ 2 = 254. ✓
Watch out. Do every test. A number can pass some tests and fail others.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for each number in turn. Write down which tests pass.
- Choose the tests. Each number has its own quick test.2, 5, 10
- Do each test on 6,000. Look only at the digits the test needs.
Test Working Divisible? 2 The last digit of 6,000 is 0. Is it 0, 2, 4, 6 or 8? yes. yes 5 The last digit of 6,000 is 0. Is it 0 or 5? yes. yes 10 The last digit of 6,000 is 0. Is it 0? yes. yes - List the numbers that pass. 2, 5, 10
Check. Divide one to check: 6,000 ÷ 2 = 3,000. ✓
Watch out. Do every test. A number can pass some tests and fail others.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 3. It saves doing the whole division.
- Remember the test for 3. Every 10, 100, 1000 is one more than a multiple of 3 (9, 99, 999), so only the digit total matters.Add up the digits. If the total is in the 3 times table, the number is divisible by 3.
- Use the test on 51. Only look at the digits that the test needs.5 + 1 = 6. 6 is in the 3 times table (3 × 2 = 6). yes.
- Answer the question. 51 is divisible by 3: Yes
Check. Divide to check: 51 ÷ 3 = 17. No remainder. ✓
Watch out. Add all the digits, not just the last one.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 3. It saves doing the whole division.
- Remember the test for 3. Every 10, 100, 1000 is one more than a multiple of 3 (9, 99, 999), so only the digit total matters.Add up the digits. If the total is in the 3 times table, the number is divisible by 3.
- Use the test on 124. Only look at the digits that the test needs.1 + 2 + 4 = 7. 7 is not in the 3 times table (6 and 9 are either side). no.
- Answer the question. 124 is not divisible by 3: No
Check. Divide to check: 124 ÷ 3 = 41 remainder 1. There is a remainder. ✓
Watch out. Add all the digits, not just the last one.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 3. It saves doing the whole division.
- Remember the test for 3. Every 10, 100, 1000 is one more than a multiple of 3 (9, 99, 999), so only the digit total matters.Add up the digits. If the total is in the 3 times table, the number is divisible by 3.
- Use the test on 372. Only look at the digits that the test needs.3 + 7 + 2 = 12. 12 is in the 3 times table (3 × 4 = 12). yes.
- Answer the question. 372 is divisible by 3: Yes
Check. Divide to check: 372 ÷ 3 = 124. No remainder. ✓
Watch out. Add all the digits, not just the last one.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 3. It saves doing the whole division.
- Remember the test for 3. Every 10, 100, 1000 is one more than a multiple of 3 (9, 99, 999), so only the digit total matters.Add up the digits. If the total is in the 3 times table, the number is divisible by 3.
- Use the test on 1,005. Only look at the digits that the test needs.1 + 0 + 0 + 5 = 6. 6 is in the 3 times table (3 × 2 = 6). yes.
- Answer the question. 1,005 is divisible by 3: Yes
Check. Divide to check: 1,005 ÷ 3 = 335. No remainder. ✓
Watch out. Add all the digits, not just the last one.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 9. It saves doing the whole division.
- Remember the test for 9. Every 10, 100, 1000 is one more than a multiple of 9 (9, 99, 999), so only the digit total matters.Add up the digits. If the total is in the 9 times table, the number is divisible by 9.
- Use the test on 63. Only look at the digits that the test needs.6 + 3 = 9. 9 is in the 9 times table (9 × 1 = 9). yes.
- Answer the question. 63 is divisible by 9: Yes
Check. Divide to check: 63 ÷ 9 = 7. No remainder. ✓
Watch out. Add all the digits, not just the last one.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 9. It saves doing the whole division.
- Remember the test for 9. Every 10, 100, 1000 is one more than a multiple of 9 (9, 99, 999), so only the digit total matters.Add up the digits. If the total is in the 9 times table, the number is divisible by 9.
- Use the test on 117. Only look at the digits that the test needs.1 + 1 + 7 = 9. 9 is in the 9 times table (9 × 1 = 9). yes.
- Answer the question. 117 is divisible by 9: Yes
Check. Divide to check: 117 ÷ 9 = 13. No remainder. ✓
Watch out. Add all the digits, not just the last one.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 9. It saves doing the whole division.
- Remember the test for 9. Every 10, 100, 1000 is one more than a multiple of 9 (9, 99, 999), so only the digit total matters.Add up the digits. If the total is in the 9 times table, the number is divisible by 9.
- Use the test on 4,321. Only look at the digits that the test needs.4 + 3 + 2 + 1 = 10. 10 is not in the 9 times table (9 and 18 are either side). no.
- Answer the question. 4,321 is not divisible by 9: No
Check. Divide to check: 4,321 ÷ 9 = 480 remainder 1. There is a remainder. ✓
Watch out. Add all the digits, not just the last one.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 9. It saves doing the whole division.
- Remember the test for 9. Every 10, 100, 1000 is one more than a multiple of 9 (9, 99, 999), so only the digit total matters.Add up the digits. If the total is in the 9 times table, the number is divisible by 9.
- Use the test on 8,073. Only look at the digits that the test needs.8 + 0 + 7 + 3 = 18. 18 is in the 9 times table (9 × 2 = 18). yes.
- Answer the question. 8,073 is divisible by 9: Yes
Check. Divide to check: 8,073 ÷ 9 = 897. No remainder. ✓
Watch out. Add all the digits, not just the last one.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 4. It saves doing the whole division.
- Remember the test for 4. Every 100 is in the 4 times table, so only the last two digits matter.Look at the last two digits. If they make a number in the 4 times table, the number is divisible by 4. (Or halve it twice.)
- Use the test on 124. Only look at the digits that the test needs.The last two digits are 24. 24 ÷ 4 = 6. yes.
- Answer the question. 124 is divisible by 4: Yes
Check. Divide to check: 124 ÷ 4 = 31. No remainder. ✓
Watch out. Use the last two digits, not just the last one: 2,136 is divisible by 4 because 36 is.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 4. It saves doing the whole division.
- Remember the test for 4. Every 100 is in the 4 times table, so only the last two digits matter.Look at the last two digits. If they make a number in the 4 times table, the number is divisible by 4. (Or halve it twice.)
- Use the test on 350. Only look at the digits that the test needs.The last two digits are 50. 50 ÷ 4 = 12 remainder 2. no.
- Answer the question. 350 is not divisible by 4: No
Check. Divide to check: 350 ÷ 4 = 87 remainder 2. There is a remainder. ✓
Watch out. Use the last two digits, not just the last one: 2,136 is divisible by 4 because 36 is.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 4. It saves doing the whole division.
- Remember the test for 4. Every 100 is in the 4 times table, so only the last two digits matter.Look at the last two digits. If they make a number in the 4 times table, the number is divisible by 4. (Or halve it twice.)
- Use the test on 1,516. Only look at the digits that the test needs.The last two digits are 16. 16 ÷ 4 = 4. yes.
- Answer the question. 1,516 is divisible by 4: Yes
Check. Divide to check: 1,516 ÷ 4 = 379. No remainder. ✓
Watch out. Use the last two digits, not just the last one: 2,136 is divisible by 4 because 36 is.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 4. It saves doing the whole division.
- Remember the test for 4. Every 100 is in the 4 times table, so only the last two digits matter.Look at the last two digits. If they make a number in the 4 times table, the number is divisible by 4. (Or halve it twice.)
- Use the test on 2,038. Only look at the digits that the test needs.The last two digits are 38. 38 ÷ 4 = 9 remainder 2. no.
- Answer the question. 2,038 is not divisible by 4: No
Check. Divide to check: 2,038 ÷ 4 = 509 remainder 2. There is a remainder. ✓
Watch out. Use the last two digits, not just the last one: 2,136 is divisible by 4 because 36 is.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 8. It saves doing the whole division.
- Remember the test for 8. 8 is 2 × 2 × 2, so we can halve three times.Halve the number three times. If you get a whole number every time, it is divisible by 8.
- Use the test on 136. Only look at the digits that the test needs.136 ÷ 2 = 68 , 68 ÷ 2 = 34 , 34 ÷ 2 = 17. yes.
- Answer the question. 136 is divisible by 8: Yes
Check. Divide to check: 136 ÷ 8 = 17. No remainder. ✓
Watch out. Halve three times. If you can only halve twice, the number is a multiple of 4 but not of 8.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 8. It saves doing the whole division.
- Remember the test for 8. 8 is 2 × 2 × 2, so we can halve three times.Halve the number three times. If you get a whole number every time, it is divisible by 8.
- Use the test on 420. Only look at the digits that the test needs.420 ÷ 2 = 210 , 210 ÷ 2 = 105 , 105 is odd: we cannot halve it exactly. no.
- Answer the question. 420 is not divisible by 8: No
Check. Divide to check: 420 ÷ 8 = 52 remainder 4. There is a remainder. ✓
Watch out. Halve three times. If you can only halve twice, the number is a multiple of 4 but not of 8.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 8. It saves doing the whole division.
- Remember the test for 8. 8 is 2 × 2 × 2, so we can halve three times.Halve the number three times. If you get a whole number every time, it is divisible by 8.
- Use the test on 5,648. Only look at the digits that the test needs.5,648 ÷ 2 = 2,824 , 2,824 ÷ 2 = 1,412 , 1,412 ÷ 2 = 706. yes.
- Answer the question. 5,648 is divisible by 8: Yes
Check. Divide to check: 5,648 ÷ 8 = 706. No remainder. ✓
Watch out. Halve three times. If you can only halve twice, the number is a multiple of 4 but not of 8.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 8. It saves doing the whole division.
- Remember the test for 8. 8 is 2 × 2 × 2, so we can halve three times.Halve the number three times. If you get a whole number every time, it is divisible by 8.
- Use the test on 7,300. Only look at the digits that the test needs.7,300 ÷ 2 = 3,650 , 3,650 ÷ 2 = 1,825 , 1,825 is odd: we cannot halve it exactly. no.
- Answer the question. 7,300 is not divisible by 8: No
Check. Divide to check: 7,300 ÷ 8 = 912 remainder 4. There is a remainder. ✓
Watch out. Halve three times. If you can only halve twice, the number is a multiple of 4 but not of 8.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for each number in turn. Write down which tests pass.
- Choose the tests. Each number has its own quick test.25, 100
- Do each test on 450. Look only at the digits the test needs.
Test Working Divisible? 25 The last two digits are 50. Is it 00, 25, 50 or 75? yes. yes 100 The last two digits are 50. Is it 00? no. no - List the numbers that pass. 25
Check. Divide one to check: 450 ÷ 25 = 18. ✓
Watch out. Do every test. A number can pass some tests and fail others.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for each number in turn. Write down which tests pass.
- Choose the tests. Each number has its own quick test.25, 100
- Do each test on 1,375. Look only at the digits the test needs.
Test Working Divisible? 25 The last two digits are 75. Is it 00, 25, 50 or 75? yes. yes 100 The last two digits are 75. Is it 00? no. no - List the numbers that pass. 25
Check. Divide one to check: 1,375 ÷ 25 = 55. ✓
Watch out. Do every test. A number can pass some tests and fail others.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for each number in turn. Write down which tests pass.
- Choose the tests. Each number has its own quick test.25, 100
- Do each test on 3,600. Look only at the digits the test needs.
Test Working Divisible? 25 The last two digits are 00. Is it 00, 25, 50 or 75? yes. yes 100 The last two digits are 00. Is it 00? yes. yes - List the numbers that pass. 25, 100
Check. Divide one to check: 3,600 ÷ 25 = 144. ✓
Watch out. Do every test. A number can pass some tests and fail others.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for each number in turn. Write down which tests pass.
- Choose the tests. Each number has its own quick test.25, 100
- Do each test on 2,090. Look only at the digits the test needs.
Test Working Divisible? 25 The last two digits are 90. Is it 00, 25, 50 or 75? no. no 100 The last two digits are 90. Is it 00? no. no - List the numbers that pass. None of them
Check. None of the tests passed, so there is a remainder every time. ✓
Watch out. Do every test. A number can pass some tests and fail others.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 6. It saves doing the whole division.
- Remember the test for 6. 6 is 2 × 3, so the number must pass both tests.The number is divisible by 2 and by 3.
- Use the test on 42. Only look at the digits that the test needs.Test 2: The last digit of 42 is 2. Is it 0, 2, 4, 6 or 8? yes.
Test 3: 4 + 2 = 6. 6 is in the 3 times table (3 × 2 = 6). yes.
Both tests must pass: yes. - Answer the question. 42 is divisible by 6: Yes
Check. Divide to check: 42 ÷ 6 = 7. No remainder. ✓
Watch out. Both tests must pass. A number that is only divisible by 2, or only by 3, is not divisible by 6.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 6. It saves doing the whole division.
- Remember the test for 6. 6 is 2 × 3, so the number must pass both tests.The number is divisible by 2 and by 3.
- Use the test on 75. Only look at the digits that the test needs.Test 2: The last digit of 75 is 5. Is it 0, 2, 4, 6 or 8? no.
Test 3: 7 + 5 = 12. 12 is in the 3 times table (3 × 4 = 12). yes.
Both tests must pass: no. - Answer the question. 75 is not divisible by 6: No
Check. Divide to check: 75 ÷ 6 = 12 remainder 3. There is a remainder. ✓
Watch out. Both tests must pass. A number that is only divisible by 2, or only by 3, is not divisible by 6.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 6. It saves doing the whole division.
- Remember the test for 6. 6 is 2 × 3, so the number must pass both tests.The number is divisible by 2 and by 3.
- Use the test on 138. Only look at the digits that the test needs.Test 2: The last digit of 138 is 8. Is it 0, 2, 4, 6 or 8? yes.
Test 3: 1 + 3 + 8 = 12. 12 is in the 3 times table (3 × 4 = 12). yes.
Both tests must pass: yes. - Answer the question. 138 is divisible by 6: Yes
Check. Divide to check: 138 ÷ 6 = 23. No remainder. ✓
Watch out. Both tests must pass. A number that is only divisible by 2, or only by 3, is not divisible by 6.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 6. It saves doing the whole division.
- Remember the test for 6. 6 is 2 × 3, so the number must pass both tests.The number is divisible by 2 and by 3.
- Use the test on 1,014. Only look at the digits that the test needs.Test 2: The last digit of 1,014 is 4. Is it 0, 2, 4, 6 or 8? yes.
Test 3: 1 + 0 + 1 + 4 = 6. 6 is in the 3 times table (3 × 2 = 6). yes.
Both tests must pass: yes. - Answer the question. 1,014 is divisible by 6: Yes
Check. Divide to check: 1,014 ÷ 6 = 169. No remainder. ✓
Watch out. Both tests must pass. A number that is only divisible by 2, or only by 3, is not divisible by 6.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test for 3 and the test for 4 on every number. Keep the numbers that pass both.
- Test every number for 3. Add up the digits. If the total is in the 3 times table, the number is divisible by 3.See the second column.
- Test every number for 4. Look at the last two digits. If they make a number in the 4 times table, the number is divisible by 4. (Or halve it twice.)See the third column.
- Put the results in a table. A number must have two “yes” answers.
Number Divisible by 3? Divisible by 4? Both? 24 yes yes yes 30 yes no no 36 yes yes yes 40 no yes no 48 yes yes yes 54 yes no no 60 yes yes yes 100 no yes no - Write the numbers that pass both tests. 24, 36, 48, 60
Check. They are all in the 12 times table: 24 = 12 × 2, 36 = 12 × 3, 48 = 12 × 4, 60 = 12 × 5. ✓
Watch out. Do not stop after one test. The number must pass both.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. A number is divisible by 3 when its digits add up to a number in the 3 times table. Try each digit in the box.
- Add the two digits we know. The box digit gets added to these.4 + 6 = 10
- Try each digit from 0 to 9. Add it to 10 and see if the total is in the 3 times table.
Digit Digit total In the 3s? 0 4 + 0 + 6 = 10 no 1 4 + 1 + 6 = 11 no 2 4 + 2 + 6 = 12 yes 3 4 + 3 + 6 = 13 no 4 4 + 4 + 6 = 14 no 5 4 + 5 + 6 = 15 yes 6 4 + 6 + 6 = 16 no 7 4 + 7 + 6 = 17 no 8 4 + 8 + 6 = 18 yes 9 4 + 9 + 6 = 19 no - Write the digits that work. 2, 5 or 8 (the numbers are 426, 456 and 486)
Check. 426 ÷ 3 = 142, 456 ÷ 3 = 152 and 486 ÷ 3 = 162. ✓
Watch out. The digit total must be in the 3 times table: 12, 15 and 18 work. 10, 11, 13, 14, 16 and 17 do not.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 9. It saves doing the whole division.
- Remember the test for 9. Every 10, 100, 1000 is one more than a multiple of 9 (9, 99, 999), so only the digit total matters.Add up the digits. If the total is in the 9 times table, the number is divisible by 9.
- Use the test on 756. Only look at the digits that the test needs.7 + 5 + 6 = 18. 18 is in the 9 times table (9 × 2 = 18). yes.
- Answer the question. 756 is divisible by 9: Yes
Check. Divide to check: 756 ÷ 9 = 84. No remainder. ✓
Watch out. Add all the digits, not just the last one.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 4. It saves doing the whole division.
- Remember the test for 4. Every 100 is in the 4 times table, so only the last two digits matter.Look at the last two digits. If they make a number in the 4 times table, the number is divisible by 4. (Or halve it twice.)
- Use the test on 756. Only look at the digits that the test needs.The last two digits are 56. 56 ÷ 4 = 14. yes.
- Answer the question. 756 is divisible by 4: Yes
Check. Divide to check: 756 ÷ 4 = 189. No remainder. ✓
Watch out. Use the last two digits, not just the last one: 2,136 is divisible by 4 because 36 is.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Use the test of divisibility for 8. It saves doing the whole division.
- Remember the test for 8. 8 is 2 × 2 × 2, so we can halve three times.Halve the number three times. If you get a whole number every time, it is divisible by 8.
- Use the test on 756. Only look at the digits that the test needs.756 ÷ 2 = 378 , 378 ÷ 2 = 189 , 189 is odd: we cannot halve it exactly. no.
- Answer the question. 756 is not divisible by 8: No
Check. Divide to check: 756 ÷ 8 = 94 remainder 4. There is a remainder. ✓
Watch out. Halve three times. If you can only halve twice, the number is a multiple of 4 but not of 8.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. Work out the correct answer yourself first. Then compare it with the answer in the question to find the mistake.
- Work it out carefully. Do the question again, one step at a time.The last two digits are 36. 36 ÷ 4 = 9. yes.
- Compare with the answer in the question. Zainab only looked at the last digit, 6.
- Find the mistake. The test for 4 uses the last two digits, not just the last one. 36 is in the 4 times table (4 × 9 = 36).
- Write the correct answer. Yes, 2,136 is divisible by 4.
Check. 2,136 ÷ 4 = 534 exactly. ✓
Watch out. For 2, 5 and 10 you only need the last digit. For 4 you need the last two digits.
Key words. A counter-example is one example that shows a statement is false.
Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.
- Try some numbers. Test the statement with some examples.Try 18 (divisible by 9) → 18 ÷ 3 = 6 ✓
Try 63 → 63 ÷ 3 = 21 ✓
Try 108 → 108 ÷ 3 = 36 ✓ - Explain why. A reason shows what is happening for every number, not just the ones we tried.9 = 3 × 3. A number in the 9 times table is 9 × something = 3 × (3 × something), so it is in the 3 times table too.
- Decide. Always
Check. Digit sums: 18 gives 9, and 9 is in the 3 times table. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.
Key words. A counter-example is one example that shows a statement is false.
Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.
- Try some numbers. Test the statement with some examples.Try 18: divisible by 3 and by 9 ✓
Try 27: divisible by 3 and by 9 ✓ - Look for one that does not work. One example that breaks the rule is called a counter-example.Try 12: 12 ÷ 3 = 4, but 12 ÷ 9 = 1 remainder 3. So 12 is divisible by 3 but not by 9.
- Explain why. A reason shows what is happening for every number, not just the ones we tried.Every multiple of 9 is a multiple of 3, but only some multiples of 3 are multiples of 9 (every third one: 9, 18, 27…).
- Decide. Sometimes
Check. Digit sum of 12 is 3: in the 3 times table but not the 9 times table. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.
Key words. A counter-example is one example that shows a statement is false.
Method. Try some numbers that end in 6. If one is not divisible by 6, Leo is wrong.
- Try 36. It ends in 6.36 ÷ 6 = 6 ✓ so 36 is divisible by 6
- Try 16. It also ends in 6.16 is even, but 1 + 6 = 7 is not in the 3 times table. 16 ÷ 6 = 2 remainder 4. So 16 is not divisible by 6.
- Answer Leo. One example that breaks the rule (a counter-example) is enough.No. 16 ends in 6 but is not divisible by 6. (Also 12 is divisible by 6 but does not end in 6.)
Check. A number is divisible by 6 only if it passes the test for 2 and the test for 3. ✓
Watch out. Ending in 6 only tells you the number is even. It does not tell you about the 3 test.
Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.
Method. A number is divisible by 4 if its last two digits are in the 4 times table. Make every number with the cards and test the last two digits.
- Make all six numbers. Put each card first in turn.345, 354, 435, 453, 534, 543
- Look at the last two digits of each one. Use the test for 4.
Number Last two digits In the 4s? 345 45 no 354 54 no 435 35 no 453 53 no 534 34 no 543 43 no - Answer Zara. All six answers are “no”.No. None of the numbers is divisible by 4.
Check. The possible last two digits are 34, 35, 43, 45, 53 and 54. The 4 times table has 32, 36, 44, 48… and none of these match. ✓
Watch out. Check all the possible numbers, not just one or two.