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Difference between revisions of "2001 AMC 10 Problems/Problem 6"

(Created page with '== Problem == Let <math> P(n) </math> and <math> S(n) </math> denote the product and the sum, respectively, of the digits of the integer <math> n </math>. For example, <math> P(…')
 
(→‎Solution)
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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>.
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== See Also ==
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{{AMC10 box|year=2001|num-b=5|num-a=7}}

Revision as of 12:34, 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 $\boxed{\textbf{(E) }9}$.

See Also

2001 AMC 10 (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
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