document.write( "Question 1205493: Solve : x! × y! = 120 × (xy)! \n" ); document.write( "
Algebra.Com's Answer #842345 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Ikleyn has found two solutions, (x,y) = (5,0) and (x,y) = (0,5) \r\n" );
document.write( "but has failed to show that there are no other solutions.  But it is\r\n" );
document.write( "necessary to show that there are no other solutions, not just to\r\n" );
document.write( "find two solutions and verify that they are both solutions.\r\n" );
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document.write( "\"x%21%2Ay%21\"\"%22%22=%22%22\"\"120%2A%28xy%29%21\"\r\n" );
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document.write( "Those will be the only solutions if and only if\r\n" );
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document.write( "\"%28xy%29%21%2F%28x%21y%21%29\"\"%22%22%3C%3E%22%22\"\"1%2F120\", for \"%22%28x%2Cy%29%22%3C%3E%22%285%2C0%29%22\" and \"%22%28x%2Cy%29%22%3C%3E%22%280%2C5%29%22\"\r\n" );
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document.write( "Assume \"x+%3C=+y\"\r\n" );
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document.write( "If x = 0, then \"%28xy%29%21%2F%28x%21y%21%29\"\"%22%22=%22%22\"\"%280%2Ay%29%21%2F0%21y%21\"\"%22%22=%22%22\"\"1%2Fy%21\"\"%22%22%3C%3E%22%22\"\"1%2F120\", except for the solution (x,y) = (0,5)\r\n" );
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document.write( "If x = 1, then \"%28xy%29%21%2F%28x%21y%21%29\"\"%22%22=%22%22\"\"%281%2Ay%29%21%2F1%21y%21\"\"%22%22=%22%22\"\"y%21%2Fy%21\"\"%22%22=%22%22\"\"1\"\"%22%22%3C%3E%22%22\"\"1%2F120\"\r\n" );
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document.write( "If \"2%3C=x%3C=y\" then \"%28xy%29%21%2F%28x%21y%21%29\"\"%22%22=%22%22\"\"%22%22=%22%22\"\"%22%22=%22%22\"\"%28%28x%2B1%29%2A%22%22%2A%22%22%2Axy%29%2F%281%2A%22%22%2A%22%22%2Ay%29\"\r\n" );
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document.write( "The numerator of that last fraction has xy-x indicated factors and the\r\n" );
document.write( "denominator has y factors.\r\n" );
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document.write( "Since \"y%3E=2\" and \"x%3E=2\",\r\n" );
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document.write( "\"x%3E=2\"\r\n" );
document.write( "Subtract 1 from both sides\r\n" );
document.write( "\"x-1%3E=1\"\r\n" );
document.write( "Divide both sides by x-1\r\n" );
document.write( "\"1%3E=1%2F%28x-1%29\"\r\n" );
document.write( "\"y%3E=2=1%2B1%3E=1%2B1%2F%28x-1%29\"\r\n" );
document.write( "\"y%3E=1%2B1%2F%28x-1%29\"\r\n" );
document.write( "\"y%3E=%28x-1%2B1%29%2F%28x-1%29\"\r\n" );
document.write( "\"y%3E=x%2F%28x-1%29\"\r\n" );
document.write( "\"y%28x-1%29%3E=x\"\r\n" );
document.write( "\"xy-y%3E=x\"\r\n" );
document.write( "\"xy-x%3E=y\"\r\n" );
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document.write( "This proves the numerator has at least as many indicated factors as the\r\n" );
document.write( "denominator.  The first x factors of the denominator are, respectively, less \r\n" );
document.write( "than or equal to the first x factors of the numerator.  The remaining indicated\r\n" );
document.write( "factors of the numerator, if any, are even greater. Therefore, the numerator of\r\n" );
document.write( "the fraction \"%28%28x%2B1%29%2A%22%22%2A%22%22%2Axy%29%2F%281%2A%22%22%2A%22%22%2Ay%29\" is greater than the denominator, and the fraction \r\n" );
document.write( "is greater than or equal to 1, and therefore not equal to \"1%2F120\".\r\n" );
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document.write( "The case \"y%3C=x\" is proved by symmetry.\r\n" );
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document.write( "Therefore the 2 solutions Ikleyn found above are the ONLY solutions.\r\n" );
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document.write( "Edwin
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