Divisibility Testing, Part 002

As promised in my previous post, I am here to show you those tests I have learned for divisibility by 5 and 10, 4 and 8.To test for divisibility by 5, you only need to look at the units digit. Is the units digit 0 or 5? If that is the case, your number is divisible by 5. Furthermore, a number whose units digit is 5 will be an odd number when you divide by 5, and even if the units digit is 0.

A number is divisible by 10 if its units digit is zero. Moreover, when you divide a digit that is not divisible by 10, the remainder is the same as the units digit.

Testing for 4 is similar. You need to check that the last two digits fall into a specific pattern, below. The digits left of the hundreds column are irrelevant.

All you have to know here is that 20 is a multiple of 4 (5×4=20), and that multiples of 20 (such as 40, 60, 80 and 100) are also multiples of 4.

Let’s look at 156.

Here, we see that 156=100 + 56. As 100 is a multiple of 4, we can discard it, leaving the 56.

Now 56 = 40 + 16, or (2×20)+16. Discarding 40, we have the remainder of 16 – which is a multiple of 4 (4×4=16). 156 is divisible by 4.

So to check for divisibility by 4 – or rather to check that a number is a multiple of 4 – the last two digits must be a multiple of 20 + a known multiple of 4, either 04, 08, 12, 16 or 00.

Now test the following numbers to see if they are multiples of 4.


Now divisibility by 8, the last exercise in this post, is very similar. As 8 is 2×4, so we have to scale up the test by 2.

The known multiples of 8 up to 5×8 are 08, 16, 24, 32 and 40. Note how the units digit goes down by increments of 2, only to begin again at 8 with 48, 56, 64, 72 and 80 – also note how the tens digit also goes up in increments of 1 each time.

Just as 25 multiples of 4 lie between 04 and 100, so 25 multiples of 8 lie between 08 and 200 – and within that range are five cycles, each 40 higher than the last. 8, 48, 88, 128 and 168 are the start of their respective cycles. This predictability is something you can use.

To check that a number, therefore, is divisible by 8, check the three digits – hundreds, tens and units. Discard all numbers in the higher columns, then discard the known multiple of 200 from the hundreds digit (you ought to end up with a hundreds digit of either 0 or 1), and finally discard the known multiple of 40 from that remainder.

If the remainder is 00, 08, 16, 24 or 32, the number is a multiple of 8. The process takes just two steps.

Try it for yourself.


I learned these methods from the friend who introduced me to Vedic Mathematics and rekindled a love of numbers in me. By teaching me to view numbers not as divisible by smaller numbers, but as multiples of the smaller numbers, she cut out a lot of the hard work surrounding division – and banished a spectre which had bothered me since my earliest schooldays by showing me how, among other things, sixteen simple rules penned thousands of years ago really can improve mathematical ability even in the oldest people and the most set in their ways.

Imagine being able to teach this to children from the beginning. What will they be able to do when they reach my age?

Next, I will be extending the methods I learned to checking for divisibility by 11, and then look at how to check for divisibility by awkward numbers, such as 7, 13 and 17.

– The Vedic Maths Forum India