document.write( "Question 1062344: How many natural numbers x and y are there such that \"1%2Fsqrt%28x%29%2B1%2Fsqrt%28y%29+=+1%2Fsqrt%2820%29\" \n" ); document.write( "
Algebra.Com's Answer #677262 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "There are 2 solutions in integers.  [3 if you count \r\n" );
document.write( "interchanging the x and y values on one of them as \r\n" );
document.write( "a separate solution.]\r\n" );
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document.write( "\"1%2Fsqrt%28x%29%2B1%2Fsqrt%28y%29+=+1%2Fsqrt%2820%29\"\r\n" );
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document.write( "\"1%2Fsqrt%28x%29%2B1%2Fsqrt%28y%29+=+1%2F%282sqrt%285%29%29\"\r\n" );
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document.write( "Let \"x=+5a%5E2\", and \"y=5b%5E2\"\r\n" );
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document.write( "\"1%2Fsqrt%285a%5E2%29%2B1%2Fsqrt%285b%5E2%29+=+1%2F%282sqrt%285%29%29\"\r\n" );
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document.write( "\"1%2F%28a%2Asqrt%285%29%29%2B1%2F%28b%2Asqrt%285%29%29+=+1%2F%282sqrt%285%29%29\"\r\n" );
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document.write( "Multiply through by \"sqrt%285%29\"\r\n" );
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document.write( "\"1%2Fa%2B1%2Fb+=+1%2F2\"\r\n" );
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document.write( "\"2b%2B2a+=+ab\"\r\n" );
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document.write( "\"2%28b%2Ba%29=ab\"\r\n" );
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document.write( "\"2b%2B2a=ab\"\r\n" );
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document.write( "\"2a=ab-2b\"\r\n" );
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document.write( "\"2a=b%28a-2%29\"\r\n" );
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document.write( "\"%282a%29%2F%28a-2%29=b\"\r\n" );
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document.write( "\"b+=+%282a%29%2F%28a-2%29\"\r\n" );
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document.write( "The smallest integer for \"a\" that will give\r\n" );
document.write( "a positive value for b is a=3\r\n" );
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document.write( "If \"a=3\", \"b+=+%282%2A3%29%2F%283-2%29=6%2F1=6\"\r\n" );
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document.write( "If \"a=4\", \"b+=+%282%2A4%29%2F%284-2%29=8%2F2=4\"\r\n" );
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document.write( "If \"a=5\", \"b+=+%282%2A5%29%2F%285-2%29=10%2F3\"  <--not an integer.\r\n" );
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document.write( "If \"a=6\", \"b+=+%282%2A6%29%2F%286-2%29=12%2F4=3\" \r\n" );
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document.write( "That's all the solutions which are integers,\r\n" );
document.write( "because \"%282a%29%2F%28a-2%29\" is a decreasing function\r\n" );
document.write( "from b=3 approaching b=2 asymptotically.\r\n" );
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document.write( "Since \"x=+5%2Aa%5E2\", and \"y=5%2Ab%5E2\"\r\n" );
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document.write( "Substituting a=3, b=6 \"x=+5%2A3%5E2=5%2A9=45\", and \"y=5%2A6%5E2=5%2A36=180\"\r\n" );
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document.write( "Substituting a=4, b=4 \"x=+5%2A4%5E2=5%2A16=80\", and \"y=5%2A4%5E2=5%2A16=80\"\r\n" );
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document.write( "Substituting a=6, b=3 \"x=+5%2A6%5E2=5%2A36=180\", and \"y=5%2A3%5E2=5%2A9=45\"\r\n" );
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document.write( "There are 2 solutions, (x,y) = (45,180), (x,y) = (80,80)\r\n" );
document.write( "There are 3 if you count (x,y) = (180,45) as a separate solution\r\n" );
document.write( "from the first one.\r\n" );
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document.write( "Checking:\r\n" );
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document.write( "1. x = 45, y = 180\r\n" );
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document.write( "\"1%2Fsqrt%2845%29%2B1%2Fsqrt%28180%29+=+1%2Fsqrt%2820%29\"\r\n" );
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document.write( "\"1%2F%283sqrt%285%29%29%2B1%2F%286sqrt%285%29%29+=+1%2F%282sqrt%285%29%29\"\r\n" );
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document.write( "\"2%2F%28+6sqrt%285%29+%29%2B1%2F%28+6sqrt%28180%29+%29+=+1%2F%28+2sqrt%285%29+%29\"\r\n" );
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document.write( "\"3%2F%286sqrt%285%29%29+=+1%2F%282sqrt%285%29%29\"\r\n" );
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document.write( "\"1%2F%282sqrt%285%29%29+=+1%2F%282sqrt%285%29%29\"\r\n" );
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document.write( "That shows that it is a solution.\r\n" );
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document.write( "x = 80, y = 80\r\n" );
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document.write( "\"1%2Fsqrt%2880%29%2B1%2Fsqrt%2880%29+=+1%2Fsqrt%2820%29\"\r\n" );
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document.write( "\"2%2Fsqrt%2880%29+=+1%2Fsqrt%2820%29\"\r\n" );
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document.write( "\"2%2F%284sqrt%285%29%29+=+1%2F%282sqrt%285%29%29\"\r\n" );
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document.write( "\"1%2F%282sqrt%285%29%29+=+1%2F%282sqrt%285%29%29\"\r\n" );
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document.write( "That shows that it is also a solution.\r\n" );
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document.write( "Edwin
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