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2001 AMC 12 Problem 2

Problem 2 of 25EasierNumber 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. Since a≠0,a \neq 0, we can divide by aa to get b=9.b = 9. The units digit of NN is 9.9. Thus, the correct answer is E.

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Concepts: digits · place value

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