Ever had to factor quadratics? You need to find factors to be able to do it. Ask any grade 10 student!
Or how about reducing fractions (or ratios)? How do we find out if we can reduce the fraction 123/78 ? We need to do that sort of thing in math class!
There are a few tricks.
Let’s try checking if 3 is a factor. It’s a trick people are less likely to know, and it’s easy to do: Take a really big number. Add up all the digits. Can you divide that sum by 3? If so, the really big number can also be divided by 3!
So 123/78. Can we reduce it?
First let’s check 123: 1+2+3=6. 3 is a factor of 6. That means that 3 is a factor of 123 also.
Now let’s check 78: 7+8 = 15. 3 is a factor of 15 so 3 is a factor of 78.
So, we can reduce 123/78 by dividing top and bottom by 3. Now we need to do the division by hand, in our head, or if really necessary, on a calculator!
123/3=41;
78/3=26;
Therefore123/78 = 41/26
Shall we try a really big number? 354296505: 3+5+4+2+9+6+5+0+5= 39; Hmm still big-ish; We can do the same trick on 39 now. 3+9=12 – now THAT’s divisible by 3. So 39 is too, and 354296505 is too!
But we can make it easier on ourselves. We can ignore all the 3’s, 6’s and 9’s (and 0’s) in the big number. [Why? If we add a multiple of 3 (like 3, or 6, or 9) to a multiple of 3, it’s still a multiple of 3.] So let’s try 354296505 again, ignoring 3’s, 6’s, and 9’s: 5+4+2+5+5 = 21 – and 21 is divisible by 3 [and 21 plus the 3’s, 6’s and 9’s must also be a multiple of 3] and therefore the big number, 354296505 is divisible by 3.
The same trick works for checking if a number is divisible by 9. Add up the digits and see if that sum can be divided by 9. If it can, the big number is also divisible by 9.
One note of clarification, if needed: I am using “divisible by”, a “multiple of” and “can be divided by” interchangeably. They all mean the same thing. And reversing the numbers, “is a factor of”: 20 is divisible by 5; or 5 is a factor of 20.
Here are a few other tricks, some of which you may already know.
| Number | How to know if it is a factor | Easy ways to do the division (for a few of them) | Examples |
| 2 | If it’s even, it’s divisible by 2 (The even numbers end in 0, 2, 4, 6, or 8) | 95376: ends in 6, which is an even number, so is divisible by 2. | |
| 3 | add up the digits of the number; if the sum is divisible by 3, the original number is also divisible by 3 | 76431: 7+6+4+3+1 = 21; 21 is divisible by 3 therefore 3 is a factor of 76431. 76430: 7+6+4+3+0 = 20 3 is NOT a factor of 20, therefore it is not a factor of 76430 | |
| 4 | Ignore all but the final 2 digits; subtract 20, 40, 60 or 80 to bring it to a lower number; if this number is divisible by 4, the complete number is also divisible by 4; | 238594: look at the last 2 digits: 94. Subtract 80; 94 - 80 = 14; 14 is NOT divisible by 4 so 238594 is not either 7608372: 72 – 60 = 12; 12 is divisible by 4 so 7608372 is also. | |
| 5 | If it ends in a “5” or a “0” it is divisible by 5 | Double the number (or add it to itself); then get rid of the last 0. | 115 / 5 = (115 + 115) / 10 = 230 / 10 = 23 |
| 6 | if the number if both even (divisible by 2) and divisible by 3 (see above) then it is divisible by 6 | 54270: 54270 ends in “0” therefore is even; 5+4+2+7+0= 18; 18 is divisible by 3 therefore 3 is a factor of 54270; even AND divisible by 3 = divisible by 6 | |
| 9 | add up the digits of the number; if the sum is divisible by 9, the original number is also divisible by 9 | 54270: 5+4+2+7+0= 18; 18 is divisible by 9 therefore 9 is a factor of 54270; 76431: 7+6+4+3+1 = 21; 21 is NOT divisible by 9 therefore 9 is a factor of 76431; | |
| 10 | If it ends in a “0” it is divisible by 10 | Get rid of the last “0” | 120 / 10 = 12 23400 / 10 = 2340 |
Have fun impressing your friends with your wizardry!