**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 .

It is a good problem solve