N is the smallest 5-digit number which when divided by 2, 2^{2}, 2^{3}, 2^{4}, \dots, 2^{n} leaves a remainder 1. What is the value of n?

Explanation

If N leaves a remainder of 1, then N-1 is a multiple of 2^n. The smallest 5-digit number is 10000. 2^{13} = 8192 and 2^{14} = 16384. For n=14, the smallest valid N over 10000 is 16384+1=16385, making 14 the largest possible valid exponent that distinctly sets this minimum.

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