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Problem

Consider the following results.

10¹ + 1 = 11
10² 1 = 99 = 9 11
10³ + 1 = 1001 = 91 11
10⁴ 1 = 9999 = 909 11
10⁵ + 1 = 100001 = 9091 11

Prove that 10ⁿ1 is divisible by 11 if n is even and 10ⁿ+1 is divisible by 11 if n is odd.