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Tests of Divisibility of Numbers

We have already learnt the divisibility test for a number by 2, 3, 5, 9, and 10. Now, we will discuss
the divisibility test for the number of a kind abc and abcd, i.e., the number of the form 100a + 10b + c
and 1000a + 100b + 10c + d. Now, we will analyse the divisibility rules and see how they work.
Divisibility Test for 2

We know that a number is divisible by 2, if its units digit is 0, 2, 4, 6 or 8.
For example, the numbers 100, 502, 404, 700, 1008, etc., are divisible by 2.

Let us write a 3-digit number abc in general form.
abc = 100a + 10b + c = 2 × 50a + 2 × 5b + c = 2 × (50a + 5b) + c = 2x + c, where x = 50a + 5b.

Clearly, ‘abc’ will be divisible by ‘2’ if and only if ‘c’ is divisible by 2.
Similarly, write a 4-digit number abcd in general form.

abcd = 1000a + 100b + 10c + d = 2(500a + 50b + 5c) + d = 2x + d, where x = (500a + 50b + 5c)
So, abcd will be divisible by 2 if its units digit, i.e., d is an even
number, i.e., 0, 2, 4, 6, or 8. Hence we say that, Quick Check

Check the divisibility of
A number is divisible by 2, if units digit of the number is an even the following numbers
number i.e., 0, 2, 4, 6 and 8. by 2:

Divisibility Test for 10 1. 1080 2. 9998
3. 10887
4. 555550
We know that a number is divisible by 10 if its units digit is 0.

For example, 30, 6710, 1100, 8000, etc. are numbers whose units digit is 0, so each of these number
is divisible by 10.
The general form of a 3-digit number abc is,

abc = 100a + 10b + c = 10 × (10a + b) + c = 10x + c, where x = 10a + b.
Clearly ‘abc’ will be divisible by ‘10’ if and only if c = 0.

Similarly, write a 4-digit number abcd in general form.
abcd = 1000a + 100b + 10c + d = 10(100a + 10b + c) + d = 10x + d, where x = (100a + 10b + c)

So, abcd will be divisible by ‘10’ if its units digit, d = 0.

Thus, we can conclude that, if the units digit of a number is 0, then the number is divisible by 10.

Divisibility Test for 5

We know that a number is divisible by 5 if its units digit is either 5 or 0.
For example, 35, 675, 1100, 8000, etc. are numbers whose units digits are either 5 or 0, so each of
these number is divisible by 5.

The general form of a 3-digit number abc is,

abc = 100a + 10b + c = 5 × (20a + 2b) + c = 5x + c, where x = 20a + 2b.

347 Playing With Numbers