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For positive integer \(n\), define \(M_n\) as an \(n\)-digit positive integer such that in decimal representation, the last \(n\) (rightmost) digits of \(M_n^2\) is \(M_n\) itself. Denote \(L_n\) as the sum of all possible values of \(M_n\). Find the 14th smallest integer \(k\) that doesn't satisfy the equation \(L_k=10^k+1\).

Details and Assumptions:

  • As an explicit example: If \(n=3\), then \(M_3 = 376,625\) because \(376^2=141\underline{376}\) and \(625^2=390\underline{625}\) and then \(L_3=10^3+1 \) but if \(n=4\), \(M_4=9376\) only, so \(L_4\ne10^4+1\).

  • I don't consider \(k=1 \) as the smallest integer \(k\) that doesn't satisfy the equation \(L_k = 10^k+1\).

Inspiration: This question that inspired this question.
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