Test of Mathematics Solution Subjective 32 (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.

Also see: Cheenta I.S.I. & C.M.I. Entrance Course

## Problem

Show that the number 11…1 with digits is divisible by

## Solution

We use induction. For n=1; we check 111 is divisible by 3. Assuming that the result host for n=k, we establish that it holds for n=k+1.

The number 111…111 (with digits) can be written in 3 blocks each having 1’s. Hence we can write it as where {111…111} denotes 1’s.

Taking {111…111} common we have . By induction (111…111) is divisible by and we also have 3 dividing as it’s sum of digits is 3 (it has only three 1’s and rest are 0) .

Hence (111…111) having 1’s is divisible by .

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