document.write( "Question 1145557: When 125 is added to a number, it becomes a perfect square. When 176 is added to that same number, it also becomes a perfect square. What is the number? \n" ); document.write( "

Algebra.Com's Answer #766782 by ikleyn(52794) $\"\"$ $\"About$

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document.write( "You are given\r\n" );
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document.write( "    x + 125 = n^2     (1)\r\n" );
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document.write( "    X + 176 = m^2     (2)\r\n" );
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document.write( "Subtract (1) from (2).  You will get\r\n" );
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document.write( "    m^2 - n^2 = 176 - 125,   or\r\n" );
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document.write( "    m^2 - n^2 = 51.\r\n" );
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document.write( "It implies\r\n" );
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document.write( "    (m+n)*(m-n) = 51.     (*)\r\n" );
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document.write( "          It is my basic equation, and I am going to find all its integer solutions.\r\n" );
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document.write( "          Notice, that if the pair of positive numbers (m,n) is the solution, then 3 other pairs \r\n" );
document.write( "          (-m,-n), (-m,n) and (m,-n) are the solutions, too.\r\n" );
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document.write( "          Therefore, I will look and search for positive solution pairs (m,n) only, keeping in mind that \r\n" );
document.write( "          every such a pair brings 3 other solutions, playing with their signs.\r\n" );
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document.write( "          But since my \"x\" depends only on m^2 and n^2, this playing with signs does not matter, at all.\r\n" );
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document.write( "Since \"m\" and \"n\" are integer numbers, it implies that \r\n" );
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document.write( "EITHER\r\n" );
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document.write( "        m + n = 51,     (3)\r\n" );
document.write( "        m - n =  1,     (4)\r\n" );
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document.write( "OR\r\n" );
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document.write( "        m + n = 17,     (5)\r\n" );
document.write( "        m - n =  3.     (6)\r\n" );
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document.write( "From equations (3), (4), the solution is  m= 26, n= 25.\r\n" );
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document.write( "From equations (5), (6), the solution is  m= 10, n=  7.\r\n" );
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document.write( "Thus EITHER\r\n" );
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document.write( "        x + 125 = n^2 = 25^2 = 625, which implies  x= 625-125 = 500,\r\n" );
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document.write( "     OR\r\n" );
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document.write( "        x + 125 = n^2 = 7^2 = 49,  which implies  x= 49-125 = -76.\r\n" );
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document.write( "ANSWER.  The problem has two solutions .  The number under the question is EITHER 500  OR  -76.\r\n" );
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