# Divisibility by 7.

Only 206 out of 467 participants in the Juniors age group during the 3^{rd} MATH-Inic Vedic Mathematics National Challenge were able to get the correct answer to this question.

Although a simple divisibility rule for 7 is taught in grade 4, many students and even teachers often forget about it. We will use this example to establish rules which we can apply not only for 7 but for all other divisors.

Rule #1 on divisibility is: adding or removing zeroes at the end of a number does not affect its divisibility by a divisor. Since 14 is evenly divisible by 7, seventy, seven hundred, seven thousand and even 7 million are also exactly divisible by 7.

We can now explain what is meant by “divisibility by 7 (by means of proximity to a multiple of 7)” which is one of the topics included in the Level I assessment of the IAVM.

133 is divisible by 7 since 133 + 7 = 140, which is clearly divisible by 7.

216 is not divisible by 7 since 6 is the difference between 216 and 210, which is 7 x 30.

From the examples above, we can now state Rule #2 on divisibility: Adding or subtracting a multiple of the divisor to a number does not affect its divisibility by that divisor.

The operating Sutra in both rules is “By Addition and By Subtraction”.

By the above divisibility rules, If we add two multiples of 7 like 3500 and 63, we will have 3563 which is also divisible by 7.

Conversely, if we partition 3563 into 35|63 by number splitting, we can see that since both 35 and 63 are divisible by 7, then 3563 is divisible by 7.

Using the same method, we can say that 4214(42|14) is divisible by 7 while 2855(28|55) is not.

Applying this technique to our featured question, we can easily get the answer:

- 566314210 → 56 | 63 | 14 | 210
- 491423549 → 49 | 14 |
**2**| 35 | 49 - 426314217 → 42 | 63 | 14 | 21 | 7
- 351428567 → 35 | 14 | 28 | 56 | 7
- 287423549 → 28 | 7 | 42 | 35 | 49

After partitioning the choices into multiples of 7, we see that we have a **2 **which is not a multiple of 7in** option b. **So, **b) 491423549** is not divisible by 7.