Question

how to know that given number is divisibility of 11 or not

Answer 1

It is easy to tell that the following are multiples of 11: 22, 33, 44, 55, etc. But how about: 2728, or 31415? Are they divisible by 11?

Here an easy way to test for divisibility by 11. Take the alternating sum of the digits in the number, read from left to right. If that is divisible by 11, so is the original number.

So, for instance, 2728 has alternating sum of digits 2-7+2-8 = -11. Since -11 is divisible by 11, so is 2728.

Similarly, for 31415, the alternating sum of digits is 3-1+4-1+5 = 10. This is not divisible by 11, so neither is 31415.

This curious fact can be easily shown using modular arithmetic. Since 10n is congruent to (-1)n mod 11, we see that 1, 100, 10000, 1000000, etc. have remainders 1 when divided by 11, and 10, 1000, 10000, etc. have remainders (-1) when divided by 11. Thus

2728= 2*1000+7*100+2*10+8,

so its remainder when divided by 11 is just 2(-1)+7(1)+2(-1)+8(1), the alternating sum of the digits. (It's sum is the negative of what we found above because the alternation here begins with a -1.) But either way, if this alternating sum is divisible by 11, then so is the original number.