Divisibility

Here are a few tricks you can use to test the divisibility of an integer. All integers are divisible by 1 and themselves, so we'll skip this. By definition every even integer is divisible by 2. Even integers end with 0, 2, 4, 6 and 8. All even numbers are divisible by 2.

What about 3? An integer is divisible by 3 if the sum of its digits is divisible by 3. Take 36. The sum of the digits of 36 is 3 plus 6 - 9 - and 9 is 3 times 3. So 36 is divisible by 3.

36 divided by 3 is 12. What about 37? The sum of the digits is 3 plus 7, which equals 10. This isn't divisible by 3, so 37 isn't divisible by 3 either. ​ Let's move on. What integers can be divided by 4?

Well, it's the integers whose last two digits divisible by 4. Take 2016. The last two digits make the number 16, and 16 is divisible by 4 evenly. 16 divided by 4 is 4, so 2016 is also evenly divisible by 4. Here we're not looking at the sum of the digits, but at the number formed by the last two digits.

What about 5? This is easy. All numbers whose last number is 0 or 5 are evenly divisible by 5. 5, 10, 15, 20, 25, 1,000,020, 1,000,025, 1,000,030. All numbers ending with 0 or 5.

Onto the next divisor. Numbers are divisible by 6 if they are divisible by both 2 and 3. So if the sum of the digits of an even number is divisible by 3 this number is also divisible by 6. ​ 7 is a bit harder. Take the number of tens by removing the rightmost digit. From this number subtract twice the amount of ones.

Take 392. That's 39 tens. Subtract twice the number of ones. 2 times 2 is 4. 39 minus 4 is 35.

35 is evenly divisible by 7. Then 392 is too. But it would probably be quicker to try plugging 392 divided by 7 into a calculator and see if you get an integer. ​ 8 is simpler. A number is divisible by 8 if the number formed by its last three digits is divisible by 8. Take 10,064.

The last three digits form 064 and 64 is 8 times 8. So 10,064 is evenly divisible by 8. ​ 9 is a bit like 3. Take the sum of the digits of the number. If it's divisible by 9, then the number is divisible by 9. Take 198.

The sum of the digits is 1 plus 9 plus 8 - 18 - and 18 is divisible by 9 since 9 times 2 is 18. So 198 is also divisible by 9. ​ 10 is the simplest case. A number is divisible by 10 if it ends with a 0.