 # Divisibility Test by Number Splitting

Which number is not exactly divisible by 7?

1. 566314210;  b) 491423549;  c) 426314217;  d) 351428567;  e) 287423549

Only about 2 out of 5 participants in the junior category of the 3rd MATH-Inic Vedic Mathematics National Challenge held online last April 10, 2023 were able to get the correct answer to this very easy question.

Most of the participants probably applied the divisibility test for 7 taught in grade school while many others most likely tried to actually divide the choices by 7.

In Vedic Mathematics, the basis for many divisibility tests is the sutra By Addition and By Subtraction. In determining whether a number 12345678 is divisible by 4, for example, we consider only the last two digits of the number, 78. Why?

Because 100 (or all numbers ending in at least 2 zeroes) is a multiple 4 and and adding or subtracting any multiple of a divisor to a number does not affect its divisibility by that number.

Now in the example given, By Subtraction of 12345600 from 12345678, we will get only 78. Now one does not need to actually divide 78 to know if it is divisible by 4. By Addition of 4 to 78 results in 82. Since 80 is divisible by 4, 82 is not. (82-80 = 2).

In choice a,  we can repeatedly apply the sutra to reduce 566,314,210  to a number which we can immediately identify as divisible by 7:

566, 314, 210 – 560,000,000 = 6,314,210

6,314,210 – 6,300,000 = 14, 210

14,210 – 14,000 = 210

Since it is now clear that 210 is divisible by 7, we can say that 566,314, 210 is divisible by 7.

The process above can be simplified by number splitting: Choice a can be easily split into 56| 63 | 14 |210. We can easily see that 56, 63, 14 and 210 are all evenly divisible by 7.

In choice b, 491423549, we have 49 | 14 | 2 | 35 | 49. While all the other partitions are divisible by 7, the middle one, 2, is not. Therefore, this number is not divisible by 7.

Let us check the other options:

c can be split into 42 | 63 | 14 | 21 | 7

d into 35 |  14 | 28 | 56 | 7

e into 28 | 7 | 42 | 35 | 49

We can see that in all choices the numbers can be partitioned into multiples of 7. So we can sat that the numbers in c, d and e are all divisible by 7.

How about 763285593? Is it divisible by 7? By number splitting we have 7 | 63 |28 | 55 | 93  which at first do not seem to be divisible by 7. But By Addition of 7, the last four digits become 5600. Since 763285600 is divisible by 7, so is 763285593.