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 #677251 by ikleyn(52780)\"\" \"About 
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\n" ); document.write( "How many natural numbers x and y are there such that \"1%2Fsqrt%28x%29%2B1%2Fsqrt%28y%29+=+1%2Fsqrt%2820%29\"
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\n" ); document.write( "\n" ); document.write( "Answer. How many ? - Zero.\r
\n" ); document.write( "\n" ); document.write( "              In other words,  the given equation has no solutions in natural numbers. \r
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\n" ); document.write( "\n" ); document.write( "Proof\r
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document.write( "Let \"x\" and \"y\" be the solution. Then   (after multiplication both sides by \"sqrt%2820%29%2Asqrt%28x%29%2Asqrt%28y%29\" ) you will get\r\n" );
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document.write( "\"sqrt%2820%29%2A%28sqrt%28x%29+%2B+sqrt%28y%29%29\" = \"sqrt%28xy%29\"   --->       ( Divide both sides by \"sqrt%28x%29\" )\r\n" );
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document.write( "\"sqrt%2820%29%2A%281+%2B+sqrt%28y%2Fx%29%29\" = \"sqrt%28y%2Fx%29\",   or         ( after introducing the new variable t = \"sqrt%28y%2Fx%29\" )\r\n" );
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document.write( "\"sqrt%2820%29%2A%281+%2B+t%29\" = t   --->   \"sqrt%2820%29+%2B+sqrt%2820%29%2At\" = t   --->  \"sqrt%2820%29\" = \"%281-sqrt%2820%29%29%2At\"  --->  t = \"-sqrt%2820%29%2F%28sqrt%2820%29-1%29\"  --->  t = \"-%28sqrt%2820%29%2A%28sqrt%2820%29%2B1%29%29%2F%2820-1%29\"  --->  t = \"-%2820%2Bsqrt%2820%29%29%2F19\".\r\n" );
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document.write( "Then from one side, \"t%5E2\" = \"y%2Fx\" is a rational number.\r\n" );
document.write( "From the other side,  \"%28-%2820%2Bsqrt%2820%29%29%2F19%29%5E2\" is an irrational number.\r\n" );
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document.write( "Contradiction.\r\n" );
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document.write( "The contradiction proves the original statement.\r\n" );
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document.write( "The proof is completed.\r\n" );
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\n" ); document.write( "Comment from student: Thanks for sparing time for me. But one step of your answer is wrong. \"sqrt%28xy%29%2Fsqrt%28x%29=sqrt%28y%29+\"
\n" ); document.write( "But you accidentally wrote it as \"sqrt%28y%2Fx%29\" That was the part you missed. Anyway thank you.
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\n" ); document.write( "\n" ); document.write( "My response. Thank you for your feedback. You are right, I was wrong. My mistake.\r
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\n" ); document.write( "\n" ); document.write( "I just saw it from the solution by Edwin.\r
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\n" ); document.write( "\n" ); document.write( "Thank you again, and thanks to Edwin for creating right solution, which is perfect.\r
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