This is a Test of Mathematics Solution (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

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If k is an odd positive integer, prove that for any integer is divisible by

We write the given expression in two ways:

This implies

Since k is odd, we know , that is divides

Applying this to we have (n+1) divides , divides and so on. Hence we can take n+1 common from each bracket, leading us to the following expression:

. This implies divides S if n+1 is even, other wise n+1 divides S.

Now we show n (or n/2) divides S (when is odd or even respectively). To show this we write

S=

Again

Since k is odd n divides for all a from 1 to n. Hence we can take n as common and have:

. This implies divides S if n is even, other wise n divides S.

Now gcd of (n, n+1) = 1. Hence divides S.

**Proved**