Difference between revisions of "2005 AMC 10B Problems/Problem 24"
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== Problem == | == Problem == | ||
+ | Let <math>x</math> and <math>y</math> be two-digit integers such that <math>y</math> is obtained by reversing the digits | ||
+ | of <math>x</math>. The integers <math>x</math> and <math>y</math> satisfy <math>x^2 - y^2 = m^2</math> for some positive integer <math>m</math>. | ||
+ | What is <math>x + y + m</math>? | ||
+ | |||
+ | <math>\mathrm{(A)} 88 \qquad \mathrm{(B)} 112 \qquad \mathrm{(C)} 116 \qquad \mathrm{(D)} 144 \qquad \mathrm{(E)} 154 </math> | ||
+ | |||
== Solution == | == Solution == | ||
+ | Let <math>x = 10a+b, y = 10b+a</math>, [[without loss of generality]] with <math>a>b</math>. Then <math>x^2 - y^2 = (x-y)(x+y) = (9a - 9b)(11a + 11b) = 99(a-b)(a+b) = m^2</math>. It follows that <math>11|(a-b)(a+b)</math>, but <math>a-b < 10</math> so <math>11|a+b \Longrightarrow a+b=11</math>. Then we have <math>33^2(a-b) = m^2</math>. Thus <math>a-b</math> is a perfect square. Also, since <math>a-b</math> and <math>a+b</math> have the same parity, so <math>a-b</math> is a one-digit odd perfect square, namely <math>1</math> or <math>9</math>. The latter case gives <math>(a,b) = (10,1)</math>, which does not work. The former case gives <math>(a,b) = (6,5)</math>, which works, and we have <math>x+y+m = 65 + 56 + 33 = 154\ \mathbf{(E)}</math>. | ||
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== See Also == | == See Also == | ||
− | + | {{AMC10 box|year=2005|ab=B|n-b=23|n-a=25}} | |
+ | |||
+ | [[Category:Introductory Number Theory Problems]] |
Revision as of 18:52, 2 January 2009
Problem
Let and be two-digit integers such that is obtained by reversing the digits of . The integers and satisfy for some positive integer . What is ?
Solution
Let , without loss of generality with . Then . It follows that , but so . Then we have . Thus is a perfect square. Also, since and have the same parity, so is a one-digit odd perfect square, namely or . The latter case gives , which does not work. The former case gives , which works, and we have .
See Also
2005 AMC 10B (Problems • Answer Key • Resources) | ||
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