2001 AMC 10 Problems/Problem 6

Problem

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 $N$ ?

$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

Solution

Let $N=10a+b$ . We want to find a number such that $ab+a+b=10a+b$ .

If we subtract the quantity $a+b$ from both sides, we are left with

$ab=9a$ .

We can also divide both sides by a and we are left with

the units digits $b$ is $\boxed{\textbf{(E) }9}$ .

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2001 AMC 10 Problems/Problem 6

Problem

Solution