I am thinking of a fifty-digit number divisible by 239, of which, each digit is the same, except the ones digit. What is the ones digit?
(In reply to Puzzle Solution
by K Sengupta)
It is already known in terms of the last post that f(7) is divisible by 239.
For n=2, we have f(14) = (10^7)* f(7) + f(7) = (10^7 + 1)*f(7), which is divisible by 239, since 239 divides f(7).
Let the result hold true for n=t. we will show that the result holds for n=t+1.
So, f(t) is divisible by 239 by assumption.
Then, f(t+1) = (10^7)* f(t) + f(t) = (10^7 + 1)*f(t), so that 239 must divide f(t+1).
Thus, f(7n) is divisible for any given positive integer n.
Hence, the proof.