Factors, Multiples and Primes
Primary 5 · Mathematics
Factors, Multiples and Primes
Lesson 1 of 3
12 min
At a shop in Nyamirambo, Gatete is stacking soda bottles. Every crate holds exactly 12 bottles.
One crate is 12 bottles. Two crates are 24. Three crates are 36. But he can never end up with exactly 30 bottles, or exactly 40 — those numbers simply do not fit into crates of 12.
The numbers Gatete can reach have a name: they are the multiples of 12. Today you will learn to spot multiples in seconds, without doing any long division at all.
Start at 12 and keep adding 12:
12, 24, 36, 48, 60, 72, 84, 96 …
That is exactly the same as 12 × 1, 12 × 2, 12 × 3, 12 × 4, and on and on.
A multiple of a number is what you get when you multiply that number by 1, 2, 3, 4 … Multiples never run out — the list goes on forever. And notice that the very first multiple of 12 is 12 itself.
If 36 is a multiple of 12, then 36 bottles fill crates of 12 perfectly, with nothing left standing on the floor. We say 36 is divisible by 12.
These three sentences all say one thing:
So when someone asks "is 36 divisible by 12?", they are really asking "is 36 in the 12 times table?"
Long division is slow. Mathematicians noticed patterns instead. Let's discover each one.
Look at 34. Can it be split into twos? 17 × 2 = 34 — yes, it splits cleanly. Now look at 35. Seventeen twos make 34, and one lonely bean is left over — it fails.
Test any number you like and you will find the whole answer is decided by the last digit.
Rule for 2: the last digit is 0, 2, 4, 6 or 8.
Count in fives: 5, 10, 15, 20, 25, 30, 35 … every single one ends in 5 or 0. Count in tens: 10, 20, 30, 40 … every single one ends in 0.
Rule for 5: the last digit is 0 or 5.
Rule for 10: the last digit is 0.
Take 51. Add its digits: 5 + 1 = 6. Six is in the 3 times table. And sure enough, 3 × 17 = 51.
Now take 52. Add its digits: 5 + 2 = 7. Seven is not in the 3 times table. And sure enough, 52 leaves 1 left over.
Rule for 3: add up all the digits. If that total is divisible by 3, the whole number is.
Take 738. Add its digits: 7 + 3 + 8 = 18. Eighteen is 9 × 2. So 738 should be divisible by 9 — and it is: 9 × 82 = 738.
Rule for 9: add up all the digits. If that total is divisible by 9, the whole number is.
Take 1,316. Break it open: 1,316 = 1,300 + 16. Now 1,300 is 13 hundreds, and one hundred is 25 fours — so hundreds are always safe for 4. That means only the 16 on the end can cause trouble, and 16 = 4 × 4. So 1,316 is divisible by 4. Check it: 1,316 ÷ 4 = 329.
Rule for 4: look only at the last two digits. If that two-digit number is divisible by 4, the whole number is.
You do not touch a pencil. You just look.
| Question | What you look at | Answer |
|---|---|---|
| Divisible by 2? | Last digit is 2, which is even | Yes |
| Divisible by 3? | 4 + 7 + 5 + 2 = 18, and 18 = 3 × 6 | Yes |
| Divisible by 4? | Last two digits are 52, and 52 = 4 × 13 | Yes |
| Divisible by 5? | Last digit is 2, not 0 or 5 | No |
| Divisible by 9? | The digits add to 18, and 18 = 9 × 2 | Yes |
| Divisible by 10? | Last digit is 2, not 0 | No |
Do not just trust me — check two of them the long way. 4,752 ÷ 9 = 528 (and 528 × 9 = 4,752). 4,752 ÷ 4 = 1,188 (and 1,188 × 4 = 4,752). The rules were right, and each one took you about five seconds.
Tip
The rules come in three families. Remember the families, not six separate facts
Bonus: any number that passes the 9 test always passes the 3 test too — because 9 is built from 3 × 3.
Work each one out, then tap Show answer to check.
Write the first six multiples of 8.
Answer: keep adding 8 — 8, 16, 24, 32, 40, 48.
Is 3,426 divisible by 3? Is it divisible by 9?
Answer: add the digits: 3 + 4 + 2 + 6 = 15. Fifteen is 3 × 5, so yes for 3 (indeed 3,426 ÷ 3 = 1,142). But 15 is not in the 9 times table, so no for 9. Passing the 3 test does not promise you the 9 test.
Is 2,548 divisible by 4?
Answer: look only at the last two digits: 48. And 48 = 4 × 12, so yes. Check: 2,548 ÷ 4 = 637.
Which of 90, 95 and 100 are divisible by 10?
Answer: 90 and 100 — they end in 0. 95 ends in 5, so it is divisible by 5 but not by 10.
Byiringiro has 4,530 Frw. Can he split it exactly into 10 equal shares? Into 2? Into 5?
Answer: the last digit is 0, so yes to all three. Ten shares of 453 Frw, or two shares of 2,265 Frw, or five shares of 906 Frw. A number ending in 0 always passes the 2, 5 and 10 tests at once.
What you learned today
Quick check
Which of these numbers is divisible by 9?
