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2001 AMC 10 Problem 6

Problem 6 of 25EasierAlgebraNumber Theory

Let P(n)P(n) and S(n)S(n) denote the product and the sum, respectively, of the digits of the integer n.n. For example, P(23)=6P(23)=6 and S(23)=5.S(23)=5. Suppose NN is a two-digit number such that N=P(N)+S(N).N=P(N)+S(N). What is the units digit of N?N?

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Solution

Write N=10a+b.N=10a+b. Then P(N)=abP(N)=ab and S(N)=a+b,S(N)=a+b, so 10a+b=ab+a+b.10a+b=ab+a+b. This reduces to 9a=ab,9a=ab, and since a≠0,a\ne0, we get b=9.b=9. The units digit is 9.9. Thus, the correct answer is E.

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Concepts: digits · algebraic manipulation

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.