2001 AMC 12 Problem 2

Attempt Problem 2 of the 2001 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2001 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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

$2$

$3$

$6$

$8$

$9$

Answer: E

Concepts:digits place value

Difficulty rating: 1080

Solution:

Write $N = 10a + b.$ Then $P(N) = ab$ and $S(N) = a + b,$ so $10a + b = ab + a + b.$ This reduces to $9a = ab.$ Since $a \neq 0,$ we can divide by $a$ to get $b = 9.$

The units digit of $N$ is $9.$

Thus, the correct answer is E.

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