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== | ||
+ | |||
+ | Let <math>N = 10t + u</math>, where <math>t</math> is the tens digit and <math>u</math> is the units digit. | ||
+ | |||
+ | The condition of the problem is that <math>10t + u + 10u + t</math> is a perfect square. | ||
+ | |||
+ | Simplifying and factoring, we want <math>11(t+u)</math> to be a perfect square. | ||
+ | |||
+ | 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 == | ||
+ | {{AHSME box|year=1997|num-b=12|num-a=14}} | ||
+ | {{MAA Notice}} |
Latest revision as of 13:12, 5 July 2013
Problem
How many two-digit positive integers have the property that the sum of and the number obtained by reversing the order of the digits of is a perfect square?
Solution
Let , where is the tens digit and is the units digit.
The condition of the problem is that is a perfect square.
Simplifying and factoring, we want to be a perfect square.
Thus, must at least be a multiple of , and since and are digits, the only multiple of that works is itself.
Thus, is the first solution, and is the last solution. There are solutions in total, leading to answer .
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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