Divisibility Rules
We say 6 is divisible by 2 since when 6 is divided by 2, the remainder is 0. On the other hand, 5 is not divisible by 2 since when 5 is divided by 2, the remainder is not 0. Therefore, in order for one whole number to be divisible by another, the remainder has to be 0. Click here for a more precise definition for divisibility.
Do we always have to divide one number by another to see whether the remainder is 0 or not? The answer is no. That’s why we have divisibility rules—these are rules for dividing two whole numbers without really dividing them since we only care about whether the remainder is 0 or not.
A whole number is divisible by:
| 2 | if its last digit is divisible by 2, i.e., 0, 2, 4, 6 and 8; |
| 3 | if the sum of its digits is divisible by 3; |
| 4 | if its last two digits are divisible by 4, i.e., 00, 04, 08, 12, 16, 20 and so on; |
| 5 | if its last digit is divisible by 5, i.e., 0 and 5; |
| 6 | if it is divisible by both 2 and 3; |
| 8 | if its last three digits are divisible by 8; |
| 9 | if the sum of its digits is divisible by 9; |
| 10 | if its last digit is 0; |
| 12 | if it is divisible by both 3 and 4; |
| 16 | if its last four digits are divisible by 16; and last but not least, |
| 25 | if its last two digits are divisible by 25, i.e., 00, 25, 50 and 75. |
You may notice that the first whole number that is missing from the list above is 7. Not that we don’t have one for 7, but it is complicated (click here to see what it is). It might be faster just dividing the number by 7 out.
You may also notice that the rules for 2, 4, 8, 16 are almost exactly the same except the number of digits required to be divisible by 2. This is because all these numbers are powers of 2. For example, to see whether a number is divisible by 16, which is 24, therefore, its last four digits have to be divisible by 16. Rules similar to these are those for powers of 5: 5, 25, 125, etc. For example, to see whether a number is divisible by 125, which is 53, its last three digits have to be divisible by 125. There are, however, only 8 such three-digit combinations, can you name all of them?
I leave you here with this challenging problem (hint: there is only one answer):
A 10-digit number is formed by using all ten digits in such a way that its first digit, counting from left to right, is divisible by 1; its first two digits are divisible by 2; its first three digits are divisible by 3; and so on. What is this 10-digit number?