Divisibility

2

A number is divisible by 2 if the unit digit is 0, 2, 4, 6 or 8.

3

A number is divisible by 3 if the sum of digits is divisible by 3.

The principle is explained here:

Let the number be .

Then we have

Note that  is divisible by 3. So if the sum of digits, i.e. , id divisible by 3, the number is divisible by 3.

4

A number is divisible by 4 if the last two digits are divisible by 4.

Note that for any positive number, it can be written in the form . Since  is divisible by 4, so if b is divisible by 4, the number is divisible by 4.

5

A number is divisible by 5 if the unit digit is 0 or 5.

The reason for this is quite similar to that of 4. Try yourself!

6

A number is divisible by 6 if it is an even number divisible by 3.

This means that if the number is a common multiple of 2 and 3, it is divisible by 6. This is because 6 is the least common multiple of 2 and 3.

7

Let the number be . Then remove the last digit and subtract twice this digit from the remaining number to get the difference, i.e. the difference is . If the difference is divisible by 7, the number is then divisible by 7. If the number is still too large, this process can be repeated to obtain a smaller one.

The principle is explained as follow:

If  is divisible by 7, then we let  for some integer m.

Now  is also divisible by 7.

8

A number is divisible by 8 if the last three digits are divisible by 8.

The reason for this is similar to that of 4. Try yourself!

9

A number is divisible by 9 if the sum of digits is divisible by 9.

The reason for this is similar to that of 9. Try yourself!

10

A number is divisible by 10 if the unit digit is 0.

11

Let the number be . The sum of odd digits is  and that of even digits is . If the difference of these two sums is divisible by 11, the number is then divisible by 11.

We first consider the case when n is odd.

Then  and .

Note that

The sum of the three last brackets in the above is divisible by 11. Thus if the difference of and  is divisible by 11, the number is also divisible by 11.

As for the case when n is even, you may try it as the principle is the same.

13

Let the number be . Then remove the last digit and subtract seven times this digit from the remaining number to get the difference, i.e. the difference is . If the difference is divisible by 13, the number is then divisible by 13. If the number is still too large, this process can be repeated to obtain a smaller one.

The proof is of the same idea as that of 7. Try yourself!

25

A number is divisible by 25 if the last two digits are divisible by 25.

The reason for this is similar to that of 4. Try yourself!

50

A number is divisible by 50 if the last two digits are divisible by 50.

The reason for this is similar to that of 25. Try yourself!

Think!

How to test a number if it is divisible by 17?

Hint: using similar idea to that of testing if a number is divisible by 7.

Answer:

17:

Let the number be . Then remove the last digit and subtract five times this digit from the remaining number to get the difference, i.e. the difference is . If the difference is divisible by 17, the number is then divisible by 17. If the number is still too large, this process can be repeated to obtain a smaller one.