Multiplication and division
Tests of divisibility
Rather than doing the division, look at the number. A few quick checks tell you whether it will divide exactly.
What divisible means
A number is DIVISIBLE by another if it divides exactly, with no remainder. 50 is divisible by 5 because 50 ÷ 5 = 10 exactly. 50 is not divisible by 4, because 50 ÷ 4 = 12 remainder 2. Saying 'divisible by 2' is the same as saying '2 is a factor of it'.
The quick tests
• by 2 — the last digit is 0, 2, 4, 6 or 8 • by 3 — the digits add to a multiple of 3 • by 4 — the last TWO digits divide by 4 • by 5 — the last digit is 0 or 5 • by 10 — the last digit is 0 Each test looks at only part of the number, which is why it is so much faster than dividing.
Building bigger tests
You can combine tests that share no factors. Divisible by 6 = divisible by 2 AND by 3. So 48 is divisible by 6: it is even, and 4 + 8 = 12 which is a multiple of 3. But you cannot test for 8 by checking 2 and 4 — 2 and 4 share the factor 2, so the trick fails.
Using the tests to sort numbers
Given 48, 49, 50, 51 and 52, work out which fits each description: 50 is divisible by 5 — it is the only one ending in 0 or 5. 49 is a square number — 7 × 7. 48 is divisible by 8 — 48 ÷ 8 = 6. 52 is divisible by 4 — the last two digits, 52, divide by 4. 51 is a multiple of 3 — 5 + 1 = 6. Start with the test that has only one possible answer, then work through what is left.
Worked examples
Is 342 divisible by 4?
- Look at the last two digits: 42.
- 42 ÷ 4 = 10 remainder 2.
Answer: No — 342 is not divisible by 4
Is 632 divisible by 4?
- The last two digits are 32.
- 32 ÷ 4 = 8 exactly.
Answer: Yes — 632 is divisible by 4
Is 51 a multiple of 3?
- Add the digits: 5 + 1 = 6.
- 6 is a multiple of 3.
Answer: Yes — 51 is a multiple of 3
Is 48 divisible by 6?
- Divisible by 6 means divisible by 2 and by 3.
- 48 is even, so it passes the 2 test.
- 4 + 8 = 12, a multiple of 3.
Answer: Yes — 48 is divisible by 6
Which of 48, 49, 50, 51, 52 divides by 5?
- The test for 5 is a last digit of 0 or 5.
- Only 50 ends in 0.
Answer: 50
Where you see this in real life
- Checking whether a number of items can be shared equally without cutting any.
- Deciding if 342 people can be split into teams of 4 with none left over.
- Working out whether a bill divides evenly between 3 people.
Key words
- Divisible
- Divides exactly, with no remainder.
- Factor
- A number that divides into another exactly.
- Multiple
- The result of multiplying a number by a whole number.
- Remainder
- What is left over after dividing.
- Tests of divisibility
- Quick checks for whether one number divides another.
- Square number
- The result of multiplying a whole number by itself.