SOLUTION: show that 27 * 23^n + 17 * 10^2n is divisible by 11 for all positive integers n.

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Question 892508: show that 27 * 23^n + 17 * 10^2n is divisible by 11 for all positive integers n.

Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Proof by induction.
For n = 1: 27%2A23+%2B+17%2A10%5E2+=+2321. 2321/11 = 211.
Induction hypothesis: Let the statement be true for n = k: Let
27%2A23%5Ek+%2B+17%2A10%5E%282k%29 be divisible by 11.
To show: 27%2A23%5E%28k%2B1%29+%2B+17%2A10%5E%282%28k%2B1%29%29 is divisible by 11.
27%2A23%5E%28k%2B1%29+%2B+17%2A10%5E%282%28k%2B1%29%29
= 23%2A27%2A23%5Ek+%2B+17%2A10%5E%282k%2B2%29
= 23%2A27%2A23%5Ek+%2B+100%2A17%2A10%5E%282k%29
= %2827%2A23%5Ek+%2B+17%2A10%5E%282k%29%29%2B+22%2A27%2A23%5Ek+%2B+99%2A17%2A10%5E%282k%29
=
Since the first grouped term is divisible by 11 by the induction hypothesis, and the second grouped term has 11 as a factor, divisibility of 27%2A23%5E%28k%2B1%29+%2B+17%2A10%5E%282%28k%2B1%29%29 by 11 follows.