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2013 HMMT Algebra #10: Minimum Value of Rational Expression

Source:

February 17, 2013
HMMTmodular arithmetic

Problem Statement

Let NN be a positive integer whose decimal representation contains 1123511235 as a contiguous substring, and let kk be a positive integer such that 10k>N10^k>N. Find the minimum possible value of 10k−1gcd⁡(N,10k−1).\dfrac{10^k-1}{\gcd(N,10^k-1)}.
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