# By Mere Observation II

Our previous Math Tip deals with recognizing numbers divisible by 11. The next logical lesson should be how to determine the remainder when a number is divided by 11. This is the subject of this question, which was given the in Juniors Category (13 years and younger) of the 1^{st} International Vedic Mathematics Olympiad last Sept 15, 2021.

The divisibility rule for 11 is to take the alternating sum of the digits of the number. If the result is 0 or a multiple of eleven, then the number is divisible by 11. A source of confusion among young learners is whether to take the alternating sum from the left to right or right to left. If taken from the left, the alternating sum of the digits of 8294 is 8 -2 + 9 – 4 = 11. If taken from the right the result would be 4 – 9 + 2 – 8 = – 11. Both ways, the result is a multiple of 11.

A second method is to take the sums of the odd-placed and the even-placed digits and then take the difference between them, it would not matter if we do (8 + 9) – (2 + 4) or (2 + 4) – (8 + 9).

However, we must be careful in finding the remainder. The rule is to take the alternating sum from the right. For 4228, the remainder when divided by 11 is (8 – 2 + 2 – 4) = 4. For 4221, we have (1 – 2 + 2 – 4) = -3. When a negative sum is obtained, we add the smallest multiple of 11 which will make the remainder positive. In this case – 3 + 11 = 8.

If the second method is used, the odd and even placement of the digits is considered from the right and we always subtract the even placed from the odd placed digits. For our IVMO 2021 question, 247, 647, we have (7 + 6 + 4) – (4 + 7 + 2) = 4.

It is very easy to spot the answer in our featured example: we note that the 4s which are in 2nd and 5^{th} places and the 7s which are in the 1^{st} and 4th position will cancel each other. ** By Mere Observation, **6 – 2 = 4 is the remainder.

Practice: Find the remainder when the following numbers are divided by 11

- 1334
- 3777
- 2323
- 5678
- 12345
- 54321
- 25552
- 235,238
- 238,235
- 555,222

Answers to previous set of exercises:

- 224, 477 → divisible
- 135, 531 → divisible
- 246, 462 → not divisible
- 246, 246 → divisible
- 344, 553 → divisible
- 345, 345 → divisible
- 883, 535 → not divisible
- 863, 434 → divisible
- 348, 634 → divisible
- 733,227 → divisible