# By Mere Observation

A number is divisible by 11 if the alternating sum of the digits is 0 or a multiple of 11. This also means that the difference between the sum of the odd-placed digits and the sum of the even-placed digits is 0 or a multiple of 11. For example we know that 258, 115 since 2 – 5 + 8 – 1 + 1 – 5 = 0 or (5 + 1 +5) – (1 + 8 + 2) = 0.

However, there are certain number which are easily identifiable as exactly divisible by 11 “By Mere Observation” which is a Vedic Math Sutra meaning “just by looking”.

In choice A the initial 3-digit string 569 is reversed to 965 which then completes the number. So, each of the digits 5, 6 and 9 occupies an odd and an even position in the number.

In option B, the digits 247 are repeated so that 2, 4 and 7 also occupy an odd and an even position in the 6-digit number.

For choice C, the digits in 33 naturally cancel each other. Similarly the digits in 58 and 85 also result in zero sum.

Option E is harder to recognize. But if we add the two-digit strings 32 and 64, we will get 96 which is the reverse of the first two digits, 69.

Use “By Mere Observation” to identify which of the following is exactly divisible by 11.

- 224,477
- 135, 531
- 246, 462
- 246, 246
- 344, 553
- 345, 345
- 883, 535
- 863, 434
- 348, 634
- 733,227

Answers to previous exercises:

- 9.2 x 9.8 = 90.16
- 9 1/5 x 9 4/5 = 90 4/25
- 4’ 3” x 4’ 9” = 20 sq. ft., 27 sq in.
- 298 x 202 = 60, 196
- 12.3 x 12.7 = 156.21
- 8 2/9 x 8 7/9 = 72 14/81
- 6’ 1”x 6’ 11” = 42 sq. ft. 11 sq. in.
- 592 x 598 = 354, 016
- 1294 x 1206 = 1,560,564

10) 1989 x 1911 = 3,800,979