Is it true that for every positive integer n, the integer 1+10^(22n-11) is divisible by 11^2?

If so how do you prove it?

?2017-10-18T04:08:28Z

Favorite Answer

True

11²=121
1+10^(22n-11) = 1+10^[11(2n-1)]
prove 121|1+10^[11(2n-1)]

121| (10^11 + 1) is true
10^11 ≡ -1 mod 121
10^(11(2n-1)) ≡ -1^(2n-1) mod 121
2n-1 is odd
-1^(odd) = -1

10^(11(2n-1)) ≡ -1 mod 121
1+10^(11(2n-1)) ≡ 0 mod 121
121 | 1+10^(11(2n-1))
QED

Also proof by induction works.

10

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