Divisibility Test Examples

Is 10,277,208 divisible by 4?

A number is divisible by 4 if the last two digits of the number are divisible by 4. That's all there is to it, yo.

The last two digits of this beast are 08, and 8 is totally divisible by 4. Sure enough, 10,277,208 is super divisible by 4.

Is 8541 divisible by 9?

Let's test this bad boy. We start by adding all the digits together.

8 + 5 + 4 + 1 = 18

Now we figure out if that sum, 18, is divisible by 9. Yup, 9 goes into 18 evenly.

That means the entire number, 8541, is also divisible by 9.

Which of the following numbers are divisible by 12?

514
204
621
975

To be divisible by 12, a number must be divisible by both 3 and 4. So the sum of the digits must be divisible by 3, and the last two digits of the number must be divisible by 4. Here we go, here we go.

Start with 514. The sum of the digits is 5 + 1 + 4 = 10 (not divisible by 3), and the last two digits (14) are not divisible by 4. Nope, 514 isn't divisible by 12. Next?

How about 204? The sum of the digits is 2 + 0 + 4 = 6 (divisible by 3), the last two digits are 04, and 4 had better be divisible by 4. So yeah, the number 204 is divisible by 12.

621 doesn't work, though. The sum of the digits is 6 + 2 + 1 = 9, which is divisible by 3, but the last two digits aren't divisible by 4. It's a bust.

Last up, we have 975. The sum of the digits is divisible by 3 (9 + 7 + 5 = 21), but the last two digits are not divisible by 4.

Looks like only 204 wins the divisible-by-12 game. Tell him what he's won, Johnny!

Name all the numbers from 2 to 10 that 6120 is divisible by.

Right out the gate, we can see that the last digit of 6120 is a nice, even number (0), so it's divisible by 2.

The last two digits (20) make a number divisible by 4, so 6120 is divisible by 4. Luckily, the last three digits (120) make a number divisible by 8, so 6120 is also divisible by 8.

The sum of the digits of 6120 is 6 + 1 + 2 + 0 = 9, which is divisible by both 3 and by 9. That tells us 6120 is divisible by 3 and 9.

Since 6120 ends in a zero, it's divisible by 5 and 10.

Since 6120 is divisible by 2 and 3, it's also divisible by 6. Since it's divisible by 3 and 4, it's also divisible by 12. Man, we're on a roll.

Now let's try 7. To see if a number is divisible by lucky number 7, we double the last digit and subtract it from the rest of the number. The last digit of 6120 is zero, and zero doubled is still zero. Subtract zero from 612 to get 612 – 0 = 612. Hmm…is 612 divisible by 7? If we're not sure, we can repeat the test again.

Double the last digit of 612 to get 2 × 2 = 4, then subtract that from 61 to get 61 – 4 = 57. Nope, 57 isn't divisible by 7, which means 612 isn't divisible by 7, which means 6120 isn't divisible by 7, either. Whew.

So much for lucky 7. How about 11? That's the only number we haven't tested yet.

For 11, we do that whole alternating sign sum test with the digits of 6120.

6 – 1 + 2 – 0 = 7

Nope, 7 isn't divisible by 11, which means 6120 isn't, either.

So that makes 6120 divisible by 2, 3, 4, 5, 6, 8, 9, 10, and 12. Poor 7 and 11, but hey, they get all the Slurpees they want so they're probably cool with that.