Division by 3 and 9
In the last session on maths we introduced rules for dividing numbers by 2 and 5. Here we continue to find rules for helping us with larger number and to see if they are divisible by 3 and 9.
We will start with nine to introduce the idea of using a digit sum.
Don't worry if you don't know what that is at the moment you will soon get used to it.
Tests of divisibility
....or how can we tell if a number can be divided exactly by another
number?
What we are going to show you today are things you can and should use everyday to see if you can divide number by other numbers.
In fact these rules must be used regularly to be effective.
Dividing by 9
Example:
How can we tell if a number can be divided exactly by 9
Let us look at the nine times table:
| Times | Answer | Add the digits | Digit Sum |
| 1 x 9 | 9 | 0 + 9 | 9 |
| 2 x 9 | 18 | 1 + 8 | 9 |
| 3 x 9 | 27 | 2 + 7 | 9 |
| 4 x 9 | 36 | 3 + 6 | 9 |
| 5 x 9 | 45 | 4 + 5 | 9 |
| 6 x 9 | 54 | 5 + 4 | 9 |
| 7 x 9 | 63 | 6 + 3 | 9 |
| 8 x 9 | 72 | 7 + 2 | 9 |
| 9x 9 | 81 | 8 + 1 | 9 |
If we add the digits in the nine times table they always add up to nine
Do you think this will happen with 10 x 9 and 11 x 9?
Answer
Yes it does
10 x 9 = 90 = 9 + 0 = 9
11 x 9 = 99 = 9 + 9 = 18 Now 1 + 8 = 9.
This is the idea: we keep adding the digits until we have only one digit left. If this digit is 9 then the number is divisible by 9.
Remember
A quicker way of saying can be divided exactly by is divisible by.
Example:
Can 108 be divided exactly by 9? a quicker way of asking the same
question: is 108 divisible by 9?
Let us use this word
All numbers whose digit sum is 9 are divisible by 9
Do you understand this now?
I hope so, if not quickly re-read the last section. Then come back
Return to last section
Here is a table to try out your new found skill:
Remember, add up the digit sum and then say if the number is divisible by 9.
| Number | Digit sum | Divisible by 9 |
| 27 | 2 + 7 = 9 | yes |
| 234 | ||
| 271 | ||
| 860 | ||
| 963 | ||
| 774 | ||
| 189 | ||
| 560 |
| Number | Digit sum | Divisible by 9 |
| 27 | 2 + 7 = 9 | yes |
| 234 | 2 + 3 + 4 = 9 | yes |
| 271 | 2 + 7 + 1 = 10 = 1 + 0 = 1 | no |
| 860 | 8 + 6 + 0 = 14 = 1 + 4 = 5 | no |
| 963 | 9 + 6 + 3 = 18= 1 + 8 = 9 | yes |
| 774 | 7 + 7 + 4 = 18 = 1 + 8 = 9 | yes |
| 189 | 1 + 8 + 9= 18 = 1 + 8 = 9 | yes |
| 560 | 5 + 6 + 0 = 11 = 1 + 1 =2 | no |
This finding the digit sum is a very useful idea that we can use again and again.
Dividing by 3
Any number whose digit sum is 3, 6, or 9 is divisible by 3.
Example:
| Times | Answer | Add the digits | Digit Sum |
| 1 x 3 | 3 | 0 + 3 | 3 |
| 2 x 3 | 6 | 0 + 6 | 6 |
| 3 x 3 | 9 | 0 + 9 | 9 |
| 4 x 3 | 12 | 1 + 2 | 3 |
| 5 x 3 | 15 | 1 + 5 | 6 |
| 6 x 3 | 18 | 1 + 8 | 9 |
| 7 x 3 | 21 | 2 + 1 | 3 |
| 8 x 3 | 24 | 2 + 4 | 6 |
| 9x 3 | 27 | 2 + 7 | 9 |
If we add the digits in the three times table they always add up to 3, 6, or 9
Do you think this will happen with 10 x 3 and 11 x 3?
Answer
Yes it does
10 x 3 = 30 = 3 + 0 = 3
11 x 3 = 33 = 3 + 3 = 6
This is the idea: we keep adding the digits until we have only one digit left.
If this digit is 3, 6 or 9 then the number is divisible by 3.
To finish off here is a table with some numbers in it. See how quickly you can go through it. Why not time yourself?
I have done the first one for you.
| Number | Digit sum | Divisible by 3, 9, both, neither |
| 20 | 2 + 0 = 2 | neither |
| 234 | ||
| 275 | ||
| 861 | ||
| 965 | ||
| 0 | ||
| 159 | ||
| 567 |
| Number | Digit sum | Divisible by 3, 9, both, neither |
| 20 | 2 + 0 = 2 | neither |
| 234 | 2 + 3 + 4 = 9 | both 3 and 9 |
| 275 | 2 + 7 + 5 = 12 = 1 + 2 =3 | 3 |
| 861 | 8 + 6 + 1 = 15 = 1 + 5 = 6 | 3 |
| 965 | 9 + 6 + 5 = 19 = 1 + 9 = 10 = 1 + 0 = 1 | neither |
| 0 | 0 | neither |
| 159 | 1 + 5 + 9 = 15 = 1 + 5 = 6 | 3 |
| 567 | 5 + 6 + 7 = 18 = 1 + 8 = 9 | both 3 and 9 |
Do you see another pattern?
If it is divisible by 9 then it is divisible by 3
Because 3 x 3 = 9
Well how did you do?
Next time we will introduce rules to help you with both 4 and 8.
Good luck!