document.write( "Question 1069386: prove that for all integers a, if a^3 is even, then a is even \n" ); document.write( "
Algebra.Com's Answer #684643 by Edwin McCravy(20067)\"\" \"About 
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document.write( "An integer of the form 2n is even (where n is an integer).\r\n" );
document.write( "An integer of the form 2n-1 is odd (where n is an integer).\r\n" );
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document.write( "The proof is by contradiction:\r\n" );
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document.write( "We assume that a³ is even, and a is odd.\r\n" );
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document.write( "Then a = 2k-1 and \r\n" );
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document.write( "a³ = (2k-1)³ = (2k)³ + 3(2k)²(-1) + 3(2k)(-1)² + (-1)³ =\r\n" );
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document.write( "8k³ - 3(4k²) + 6k - 1 =\r\n" );
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document.write( "8k³ - 12k² + 6k - 1 =\r\n" );
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document.write( "Factor 2 out of the first three terms:\r\n" );
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document.write( "2(4k³ - 6k² + 3k) - 1\r\n" );
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document.write( "This is of the form 2n-1, which contradicts the\r\n" );
document.write( "assumption that a is odd.\r\n" );
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document.write( "Edwin
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