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product of the digits of positive integers

Source: Romanian Nationals RMO 2005 - grade 8, problem 2

March 31, 2005

Problem Statement

For a positive integer nn, written in decimal base, we denote by p(n)p(n) the product of its digits. a) Prove that p(n)≤np(n) \leq n; b) Find all positive integers nn such that 10p(n)=n2+4n−2005. 10p(n) = n^2+ 4n - 2005. Eugen Păltănea
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