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Problem 1

Part of 2013 CIIM

Problems(1)

CIIM 2013 Problem 1

Source:

6/9/2016
Given two natural numbers mm and nn, denote by m.n‾\overline{m.n} the number obtained by writing mm followed by nn after the decimal dot.
a) Prove that there are infinitely many natural numbers kk such that for any of them the equation m.n‾×n.m‾=k\overline{m.n} \times \overline{n.m} = k has no solution. b) Prove that there are infinitely many natural numbers kk such that for any of them the equation m.n‾×n.m‾=k\overline{m.n} \times \overline{n.m} = k has a solution.
CIIMCIIM 2013undergraduate