2001 AMC 12 Problem 2

Below is the professionally curated solution for Problem 2 of the 2001 AMC 12, 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.

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Concepts:digits place value

Difficulty rating: 1080

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 $N?$

$2$

$3$

$6$

$8$

$9$

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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