Prove that if the difference of the integers a, b is divisible by 100, then
is divisible by 10 000.
Here's one way:
If (a - b)|100 then we can say that (a - b)k = 100, where k is a positive integer.
Thus . So
<-- Singling out the first term of the summation
Now, each term in the series is divisible by at least 100 due to the factor of . But there is another 100 involved: from the , i > 0. So all terms are divisible by at least 10000.