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Difference between revisions of "2005 AMC 10B Problems/Problem 24"
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== 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>.
 
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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 +
== Solution 2 ==
 +
The first steps are the same as above.  Let <math>x = 10a+b, y = 10b+a</math>, where we know that a and b are digits (whole numbers less than 10).  Like above, we end up getting <math>(9a - 9b)(11a + 11b) = 99(a-b)(a+b) = m^2</math>.  This is where the solution diverges. 
 +
 +
We know that the left side of the equation is a perfect square because m is an integer.  If we factor 99 into its prime factors, we get <math>3^2 * 11</math>.  In order to get a perfect square on the left side, <math>(a-b)(a+b)</math> must make both prime exponents even.  Because the a and b are digits, a simple guess would be that <math>(a+b)</math> (the bigger number) equals 11 while <math>(a-b)</math> is a factor of nine (1 or 9).  The correct guesses are <math>a = 6, b = 5</math> causing <math>x = 65, y = 56,</math> and <math>m = 33</math>.  The sum of the numbers is 154 <b>(E)</b>.
  
 
== See Also ==
 
== See Also ==

Revision as of 19:58, 27 February 2010

Problem

Let $x$ and $y$ be two-digit integers such that $y$ is obtained by reversing the digits of $x$. The integers $x$ and $y$ satisfy $x^2 - y^2 = m^2$ for some positive integer $m$. What is $x + y + m$?

$\mathrm{(A)} 88 \qquad \mathrm{(B)} 112 \qquad \mathrm{(C)} 116 \qquad \mathrm{(D)} 144 \qquad \mathrm{(E)} 154$

Solution

Let $x = 10a+b, y = 10b+a$, without loss of generality with $a>b$. Then $x^2 - y^2 = (x-y)(x+y) = (9a - 9b)(11a + 11b) = 99(a-b)(a+b) = m^2$. It follows that $11|(a-b)(a+b)$, but $a-b < 10$ so $11|a+b \Longrightarrow a+b=11$. Then we have $33^2(a-b) = m^2$. Thus $a-b$ is a perfect square. Also, since $a-b$ and $a+b$ have the same parity, so $a-b$ is a one-digit odd perfect square, namely $1$ or $9$. The latter case gives $(a,b) = (10,1)$, which does not work. The former case gives $(a,b) = (6,5)$, which works, and we have $x+y+m = 65 + 56 + 33 = 154\ \mathbf{(E)}$.

Solution 2

The first steps are the same as above. Let $x = 10a+b, y = 10b+a$, where we know that a and b are digits (whole numbers less than 10). Like above, we end up getting $(9a - 9b)(11a + 11b) = 99(a-b)(a+b) = m^2$. This is where the solution diverges.

We know that the left side of the equation is a perfect square because m is an integer. If we factor 99 into its prime factors, we get $3^2 * 11$. In order to get a perfect square on the left side, $(a-b)(a+b)$ must make both prime exponents even. Because the a and b are digits, a simple guess would be that $(a+b)$ (the bigger number) equals 11 while $(a-b)$ is a factor of nine (1 or 9). The correct guesses are $a = 6, b = 5$ causing $x = 65, y = 56,$ and $m = 33$. The sum of the numbers is 154 (E).

See Also

2005 AMC 10B (Problems • Answer Key • Resources)
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All AMC 10 Problems and Solutions
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