Sometimes it can help to have some tricks up our sleeve when starting to divide a number. Thankfully, there are ways to check if a number is divisible by any of the numbers between 2 and 9 before you have to spend a bunch of time on trial and error. Remember, we don’t need a trick for 1, because every number is divisible by 1, and we don’t need one for 0, because you can’t divide by zero.
It’s important to know when these are worth using. The tricks for both 7 and 8 are complicated enough that they’re only really useful for larger numbers, and you’d get more use out of well-memorized times tables for most problems. However, the tricks for 3, 4, 5, 6, and 9 are quick and easy, and can save a lot of guessing time on division problems.
A number can be divided by 2 if: it is even, meaning it ends in 0, 2, 4, 6, or 8.
102 is divisible by 2. 103 is not.
A number can be divided by 3 if: adding up all its digits gives you a multiple of 3.
102 is divisible by three–if you add 1 + 0 + 2, you get 3. If we test it out, we see that 102 ÷ 3 = 34. Try this on any number where the digits add up to 3 or a multiple of 3, and you’ll see that it works.
A number can be divided by 4 if: the tens place is even, and the ones place is 0, 4, or 8, or if the tens place is odd and the ones place is 2 or 6.
60 has an even number in the tens place (6) and a 0 in the ones place, so it’s divisible by 4. If we test it out, we get 15. 72 has an odd number in the tens place (7) and a 2 in the ones, so it’s also divisible by 4; testing it out gets us 18. 86 has an even number in the tens place (8) and a 6 in the ones, so it’s not divisible by 4. If we test it out, we get 21 R2.
A number can be divided by 5 if: it ends with a 5 or a 0.
85 ends in a 5, so it’s divisible by 5–that gives us 17. 90 ends in a 0, so it’s divisible by 5–that gives us 18. 93 doesn’t end in 5 or 0, so if we try to divide it by 5, we get 18 R3.
A number can be divided by 6 if: its digits add up to a multiple of 3 and it is even.
The digits of 156 add up to 12 (1 + 5 + 6 = 12), and the number itself is even: if we divide it by 6, we get 26. The digits of 165 add up the same way, but the number is odd, and if we divide it by 6, we get 27 R3.
A number can be divided by 7 if: doubling the last digit and subtracting it from the rest gives you a multiple of 7. This one is a bit complicated, so pay close attention to these examples. Let’s say we want to know if 483 is divisible by 7. We start by doubling the last digit–in this case, 3–giving us 6. Then we subtract 6 from the rest of the digits (the “48” in 483), meaning we get:
48 – 6 = 42
If we remember our times tables, we know that 42 is a multiple of 7, so 483 should be divisible by 7. If you go ahead and try it, you’ll see that 483 ÷ 7 = 69. It works! What if we try it with 221? Doubling the last digit gives us 2, and subtracting that from the other digits (22) gives us 20, which is not a multiple of 7. If we try to divide 221 by 7, we get 31 R4.
This can be repeated as many times as you need. For example, what if we want to know if 2,723 is divisible by 7? Using the trick once gets us 272 – 6, which is 266, but we can’t easily tell if that’s divisible by 7, so we repeat the trick. We double the last 6 to get 12 and subtract it from 26 to 14. That is very clearly a multiple of 7, so 2,723 is divisible by 7. If we try it, we get 389.
A number can be divided by 8 if: the hundreds place is an even number (including 0) and the last two places make up a multiple of 8, or the hundreds place is an odd number, and the last two places make up a multiple of 4 and an odd number. This one is also complicated, but when dealing with numbers greater than 1,000, it can help. Let’s consider 7,112.
We see that the hundreds place is an odd number (1). The last two places make up 12, which is a multiple of 4 and an odd number (3). If we try dividing 7,112 by 8, we see that this trick works: we get 889. 2,672 has an even number in the hundreds place (6) and a multiple of 8 in the last two places (72), so it should also be divisible by 8. If we try it, we get 334.
A number can be divided by 9 if: the digits add up to a multiple of 9. So, if we try this with 243, we get 2 + 4 + 3 = 9, so it should be divisible. If we try it, we get 243 ÷ 9 = 27.