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Difference between revisions of "1997 AHSME Problems/Problem 13"
 
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<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 </math>
 
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 </math>
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==Solution==
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Let <math>N = 10t + u</math>, where <math>t</math> is the tens digit and <math>u</math> is the units digit.
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The condition of the problem is that <math>10t + u + 10u + t</math> is a perfect square.
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Simplifying and factoring, we want <math>11(t+u)</math> to be a perfect square.
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Thus, <math>t+u</math> must at least be a multiple of <math>11</math>, and since <math>t</math> and <math>u</math> are digits, the only multiple of <math>11</math> that works is <math>11</math> itself.
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Thus, <math>(t,u) = (2,9)</math> is the first solution, and <math>(t,u) = (9,2)</math> is the last solution.  There are <math>8</math> solutions in total, leading to answer <math>\boxed{E}</math>.
  
 
== See also ==
 
== See also ==
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{{AHSME box|year=1997|num-b=10|num-a=12}}
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{{AHSME box|year=1997|num-b=12|num-a=14}}
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{{MAA Notice}}

Latest revision as of 14:12, 5 July 2013

Problem

How many two-digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of is a perfect square?

$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8$

Solution

Let $N = 10t + u$, where $t$ is the tens digit and $u$ is the units digit.

The condition of the problem is that $10t + u + 10u + t$ is a perfect square.

Simplifying and factoring, we want $11(t+u)$ to be a perfect square.

Thus, $t+u$ must at least be a multiple of $11$, and since $t$ and $u$ are digits, the only multiple of $11$ that works is $11$ itself.

Thus, $(t,u) = (2,9)$ is the first solution, and $(t,u) = (9,2)$ is the last solution. There are $8$ solutions in total, leading to answer $\boxed{E}$.

See also

1997 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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