Show that a positive integer is divisible by 11 (continued in text)?

Question

Show that a positive integer is divisible by 11 if and only if the difference of the sum of its decimal digits in even-numbered positions and the sum of its decimal digits in odd-numbered positions is divisible by 11.

Answer 1

you can express any number like this
a+10b+100c+1000d+...
or
a+(11b-b)+(99b+b)+(1001c-c)...
because 11,99,1001,9999,...
are divisible by 11 then the remainder
will be:
a-b+c-d...
and if it's 0 then it means that the
number is divisible by 11
for example:
2453 is divisible by 11
because 2-4+5-3=0

Answer 2

I love this one. :)

10 is congruent to -1 modulo 11.

Hence, every odd power of 10 is also congruent to -1 modulo 11 and every even power of 10 is congruent to 1 modulo 11.

View the number as the sum of the series (kth digit * 10^k), and the proof falls right out.