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2001 AMC 10 Problems/Problem 6: Difference between revisions

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the units digits <math> b </math> is <math> \boxed{\textbf{(E) }9} </math>.
the units digits <math> b </math> is <math> \boxed{\textbf{(E) }9} </math>.
== See Also ==
{{AMC10 box|year=2001|num-b=5|num-a=7}}

Revision as of 15:06, 16 March 2011

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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