Test of Mathematics at the 10+2 Level

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 3^n digits is divisible by 3^n

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 3^{k+1} digits) can be written in 3 blocks each having 3^{k} 1’s. Hence we can write it as {111...111} \times 10^{(2 \times 3^k)} + {111...111} \times 10^{(3^k)} + {111...111} where {111…111} denotes 3^k 1’s.
Taking {111…111} common we have (111...111)(10^{(2 \cdot 3^k)} + 10^{(3^k)} + 1). By induction (111…111) is divisible by 3^k and we also have 3 dividing (10^{(2 \cdot 3^k)} + 10^{(3^k)} +1 ) as it’s sum of digits is 3 (it has only three 1’s and rest are 0) .
Hence (111…111) having 3^{(k+1)} 1’s is divisible by 3^{(k+1)} .