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# Test of Mathematics Solution Subjective 35 - Divisibility by 16  Test of Mathematics Solution Subjective 35 (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.

## Problem

(a) Prove that, for any odd integer n, when divided by always leaves remainder .

(b) Hence or otherwise show that we cannot find integers such that .

## Solution

For part (a) we consider and expand it's fourth power binomially to get Now ; if is even then is divisible by and if is odd is even and product of and is divisible by . Since is already divisible by we conclude when divided by gives as remainder.

For part (b) we note that when divided by , produces as the remainder. Each of the eight of fourth powers when divided by produces either (when is even) or (when is odd using part (a)) as remainder. Thus they can add up to at most (modulo ) hence can never be equal to (which is modulo ). Test of Mathematics Solution Subjective 35 (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.

## Problem

(a) Prove that, for any odd integer n, when divided by always leaves remainder .

(b) Hence or otherwise show that we cannot find integers such that .

## Solution

For part (a) we consider and expand it's fourth power binomially to get Now ; if is even then is divisible by and if is odd is even and product of and is divisible by . Since is already divisible by we conclude when divided by gives as remainder.

For part (b) we note that when divided by , produces as the remainder. Each of the eight of fourth powers when divided by produces either (when is even) or (when is odd using part (a)) as remainder. Thus they can add up to at most (modulo ) hence can never be equal to (which is modulo ).

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