Divisibility Testing, Part 001

I have posted on this subject on my own blog, and I thought I would bring some of the insights I have gained to share them with you readers here.

Of all the mathematical operations I was taught as a child, division was always considered the hardest to do by myself and my fellow pupils. It was not until adolescence that I began to learn how to speed up the process of division through divisibility tests.

I learned a number of very quick methods to test for divisibility from Figuring: The Joy of Numbers by Shakuntala Devi, and these simple key methods have remained with me to this day.

In this first part, I will share some of these methods with you.

Let’s begin with the simplest tests: divisibility by 2, 3, 6 and 9.

To test for divisibility by 2 (i.e. an even number), check the units digit (the digit on the right): if the units digit is 0, 2, 4, 6 or 8 the number is even. If the units digit is 1, 3, 5, 7 or 9, it is an odd number.

A number is divisible by 3 if the sum of all its digits is a multiple of 3: 3, 6, 9 or a multiple of 3.

Let’s look at an example: testing 45,327 for divisility by 3.

Summing 4+5+3+2+7 yields 21, which is a known multiple of 3. 45,327 is divisible by 3. You can test that by adding 2+1 to get 3.

Try it for yourself. Test the following numbers for divisibility by 3, now:-

123
237
4,386
7,184
66,247

Testing for divisibility by 6 is simple, once you know how to test for divisibility by both 2 and 3: a number which is even and which is also divisible by 3 is divisible by 6.

Some examples of this:

12 (1+2=3) is an even number, and is therefore divisible by 6.
24 (2+4=6) is also an even number, and is therefore divisible by 6.
39 (3+9=12, 1+2=3) is divisible by 3, but it is an odd number: 39 is not divisible by 6.
Finally 22 (2+2=4) is an even number, but it is not divisible by 3 so therefore it is not divisible by 6.

Try testing the following numbers for divisibility by 6:-

54
122
642
2,454
13,563

Finally testing for divisibility by 9 is very similar to testing for divisibility by 3 but far more specific. If the sum of the numbers adds up to 9, or a multiple of 9, then the number is divisible by 9.

For example 54 (5+4=9) is divisible by 9. 189 (1+8+9=18, 1+8=9) is likewise divisible by 9. 24 (2+4=6), on the other hand, is not divisible by 9.

Try testing the following numbers for divisibility by 9:-

18
351
4,571
11,223
23,480

And that is how to test for divisibility by 2, 3, 6 and 9. In the next part, I will show you the methods I learned to test for divisibility by 4, 5, 8 and 10.

– The Vedic Maths Forum India