B. Number · Operations: Divisibility, Decimals, Fractions, Ratios and PercentsDivisibility Rules for 3, 6 and 9
B2.2 A1About 40 minutes
Learning goals- I can use the digit sum to tell if a number is divisible by 3 or by 9.
- I can tell if a number is divisible by 6 by checking for 2 and 3.
- I can use all the divisibility rules from 2 to 10 (except 7) to test a number.
Words to know
digit sum
The total you get when you add the digits of a number. The digit sum of 471 is 4 + 7 + 1 = 12.
divisible
Divides with no remainder.
factor
A number that divides another number with no remainder.
Let's start with a magic trick
Ask someone to think of a number like 2 718. Add its digits: 2 + 7 + 1 + 8 = 18. 18 is divisible by 9, so you can say, without dividing, "Your number is divisible by 9!" And it is: 2 718 ÷ 9 = 302.
It isn't really magic. It's a divisibility rule, and today you'll find out why it works.
The rules
- 3: the digit sum is divisible by 3.
- 9: the digit sum is divisible by 9.
- 6: the number is divisible by both 2 and 3. (It is even, and its digit sum is divisible by 3.)
If the digit sum is still big, you can add its digits again. For 98 778: 9 + 8 + 7 + 7 + 8 = 39, and 3 + 9 = 12. 12 is divisible by 3 but not by 9, so 98 778 is divisible by 3 but not by 9.
Why does the digit sum work?
Look at 234. It is 2 hundreds, 3 tens and 4 ones.
- Each hundred is 99 + 1. The 99 splits evenly into groups of 9 (and of 3). 1 is left over.
- Each ten is 9 + 1. The 9 splits evenly. 1 is left over.
So from 2 hundreds, 2 are left over. From 3 tens, 3 are left over. And there are 4 ones. Altogether the leftovers are 2 + 3 + 4 = 9, which is the digit sum! The leftovers make 9, which is one more group of 9, so 234 is divisible by 9 (234 ÷ 9 = 26).
For 6: 6 = 2 × 3. A number that splits into twos and into threes with nothing left over also splits into sixes.
Worked exampleExample 1: Test for 3, 6 and 9
Is 1 452 divisible by 3? By 9? By 6?
- Digit sum: 1 + 4 + 5 + 2 = 12.
- 12 is divisible by 3, so 1 452 is divisible by 3.
- 12 is not divisible by 9, so 1 452 is not divisible by 9.
- 1 452 is even and divisible by 3, so it is divisible by 6. (1 452 ÷ 6 = 242.)
Divisible by 3 and 6, but not by 9.
Worked exampleExample 2: An odd number
Is 4 869 divisible by 3? By 9? By 6?
- Digit sum: 4 + 8 + 6 + 9 = 27. 27 is divisible by 9 (and by 3).
- So 4 869 is divisible by 9 and by 3. (4 869 ÷ 9 = 541.)
- 4 869 is odd, so it is not divisible by 2. That means it is not divisible by 6.
Divisible by 3 and 9, but not by 6.
Worked exampleExample 3: Solve a problem
252 students sign up for a winter carnival. Can they be put into equal teams of 6? Of 9?
- Digit sum: 2 + 5 + 2 = 9. So 252 is divisible by 3 and by 9.
- 252 is even and divisible by 3, so it is divisible by 6.
- 252 ÷ 6 = 42 and 252 ÷ 9 = 28.
Yes to both: 42 teams of 6, or 28 teams of 9.
| Divisible by | Rule |
|---|
| 2 | Last digit is 0, 2, 4, 6 or 8 |
| 3 | Digit sum is divisible by 3 |
| 4 | Last two digits make a number divisible by 4 |
| 5 | Last digit is 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 8 | Last three digits make a number divisible by 8 |
| 9 | Digit sum is divisible by 9 |
| 10 | Last digit is 0 |
Keep going
A divisibility rule can look hard at first. If you get stuck, try the rule on a small number you already know, like 18 or 27, to see how it works. Then go back to the big number. Making mistakes and trying again is how your brain learns.
Just for funA trick that always works: pick any whole number (not 0), multiply it by 9, then add the digits of the answer. Keep adding digits until only one is left. It will always be 9!
Hands-on activityDigit sum challenge
You need: a deck of cards with the 10s and face cards removed (aces count as 1), paper, a pencil and a calculator for checking.
- Deal 3 cards. Use them to make the largest 3-digit number you can.
- Find the digit sum. Is the number divisible by 3? By 9? By 6?
- Check with the calculator.
- Rearrange the cards. Does the digit sum change? (It doesn't! So if one arrangement is divisible by 3, every arrangement is.)
- Now try to make a number divisible by 6. Hint: which card must go last? (If none of your numbers works, explain why.)
Math talk: "Why doesn't moving the digits change the answer for 3 and 9, but it can change the answer for 2 and 6?"
Your turn
Yes. 4 + 7 + 1 = 12, and 12 is divisible by 3. (471 ÷ 3 = 157.)
No. 5 + 8 + 3 = 16, and 16 is not divisible by 3.
Yes. 7 + 3 + 8 = 18, and 18 is divisible by 9. (738 ÷ 9 = 82.)
Yes. 1 + 2 + 0 + 6 = 9. (1 206 ÷ 9 = 134.)
Yes. It is even, and 4 + 1 + 4 = 9, which is divisible by 3. (414 ÷ 6 = 69.)
No. 735 is odd, so it is not divisible by 2, and so not by 6.
No. 2 + 3 + 4 + 4 = 13, and 13 is not divisible by 3.
More practice: warm-up
1What is the digit sum of 345?
Yes. 2 + 7 = 9. (27 = 9 × 3.)
Yes. 5 + 1 = 6, and 6 is divisible by 3. (51 = 3 × 17.)
No. 6 + 4 = 10, and 10 is not divisible by 3.
Yes. 90 is even, and 9 + 0 = 9 is divisible by 3. (90 = 6 × 15.)
More practice: on your own
Yes. 5 + 1 + 1 + 2 = 9. (5 112 ÷ 9 = 568.)
Yes. It is even, and 3 + 5 + 7 + 0 = 15, which is divisible by 3. (3 570 ÷ 6 = 595.)
3Is 8 241 divisible by 3? By 9?
By 3: yes. By 9: no. 8 + 2 + 4 + 1 = 15. 15 is divisible by 3 but not by 9.
Yes. It is even, and 4 + 3 + 2 + 6 = 15, which is divisible by 3. (4 326 ÷ 6 = 721.)
5What digit goes in the blank so that 5_2 is divisible by 9?
2. The digit sum is 5 + _ + 2 = 7 + _. It must be 9 or 18. 7 + 2 = 9 works. 7 + 11 = 18 would need 11, which is not a digit. So the number is 522 (522 ÷ 9 = 58).
Challenge
1What is the smallest 4-digit number that is divisible by 9?
1 008. Its digit sum is 1 + 0 + 0 + 8 = 9. Numbers from 1 000 to 1 007 have digit sums from 1 to 8, so none of them work.
2Test 6 840 for divisibility by 2, 3, 4, 5, 6, 8, 9 and 10.
It is divisible by all of them! Ends in 0: 2, 5, 10. Digit sum 6 + 8 + 4 + 0 = 18: 3 and 9. Even and divisible by 3: 6. Last two digits 40 ÷ 4 = 10: 4. Last three digits 840 ÷ 8 = 105: 8.
3Can a number be divisible by 9 but not by 3? Explain.
No. 9 = 3 × 3, so every group of 9 is three groups of 3. Also, if the digit sum is divisible by 9, it is divisible by 3 too.
Quiz
1. Which number is divisible by 3?
2. Which number is divisible by 9?
3. A number is divisible by 6 when it is…
5. Which digit makes 7_4 divisible by 3?
6. What is the digit sum of 2 718?
Big idea: This lesson finishes B2.2: the divisibility rules for 3, 6 and 9, and the full list from 2 to 10 (Grade 6 does not include a rule for 7). The "why" for the digit sum is that every 10, 100 and 1000 is one more than a multiple of 9 (9, 99, 999).
Common difficulties:
- Using the digit sum for 6. The rule for 6 needs two checks: even and digit sum divisible by 3.
- Thinking a number ending in 3, 6 or 9 is divisible by 3. 13 and 26 are not.
- Adding the digits wrongly. Encourage writing the sum out.
How to help: Car licence plates and house numbers are great for quick digit sums. Encourage your child to keep trying when a rule feels confusing and to test it on small numbers first. That builds the A1 skills of perseverance and self-checking. Divisibility rules will help a lot in the next lesson on prime factors and in simplifying fractions.