document.write( "Question 1069384: prove by contradiction. for any integer n, n^2-2 is not divisible by 4 \n" ); document.write( "

Algebra.Com's Answer #684684 by Edwin McCravy(20054) $\"\"$ $\"About$

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document.write( "We will use the theorem:\r\n" );
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document.write( "If a perfect square is divisible by a prime p,\r\n" );
document.write( "it is also divisible by p².\r\n" );
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document.write( "Assume that when we divide n²-2 by 4\r\n" );
document.write( "we get an integer k\r\n" );
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document.write( "Therefore n² is divisible by 2.\r\n" );
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document.write( "Since n² is divisible by 2, and 2 is a prime,\r\n" );
document.write( "by the theorem n² must be divisible by 2², or 4. \r\n" );
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document.write( "Therefore 2k+1 must be divisible by 2, \r\n" );
document.write( "but 2k+1 is an odd number and is not divisible\r\n" );
document.write( "by 2, so we have reached a contradiction.\r\n" );
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document.write( "Therefore the assumption that n²-2 is divisible by 4\r\n" );
document.write( "is incorrect, and therefore n²-2 is not divisible by 4.\r\n" );
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document.write( "Edwin

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