document.write( "Question 1070276: Show that if n is odd integer, then n^3-n is a multiple of 24. \n" ); document.write( "
Algebra.Com's Answer #685408 by Edwin McCravy(20055)\"\" \"About 
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document.write( "She assumed that\r\n" );
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document.write( "\"If n is odd, then both (n-1) and (n+1) are even, i.e. are multiples\r\n" );
document.write( "of 2.\"\r\n" );
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document.write( "I agree with that without proof.  However when she says:\r\n" );
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document.write( "\"Moreover, one of these two is a multiple of 4.\"\r\n" );
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document.write( "although that is true, I think it must be proved before she can use it\r\n" );
document.write( "legitimately.\r\n" );
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document.write( "I do an induction proof:\r\n" );
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document.write( "\"n%5E3-n=n%5E2%28n-1%29=n%28n-1%29%28n%2B1%29\"\r\n" );
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document.write( "Since n is odd, let n = 2m+1 where m is any integer\r\n" );
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document.write( "Then we have to prove that:\r\n" );
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document.write( "then 8m^3+12m^2+4m}}} is a multiple of 24 for any integer m\r\n" );
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document.write( "If m=1\r\n" );
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document.write( "4(1)(1+1)(2*1+1) = 4(2)(2+1) = 4(2)(3) = 24\r\n" );
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document.write( "and 24 is a multiple of 24.\r\n" );
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document.write( "Assume that m=k is such that \"8k%5E3%2B12k%5E2%2B4k\" = 24 times some integer,\r\n" );
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document.write( "Now we consider   \"8%28k%2B1%29%5E3%2B12%28k%2B1%29%5E2%2B4%28k%2B1%29\".  When we expand that\r\n" );
document.write( "out we get \r\n" );
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document.write( "\"8k%5E3%2B36k%5E2%2B52k%2B24\"\"%22%22=%22%22\"\"%288k%5E3%2B12k%5E2%2B4k%29%2B%2824k%5E2%2B48k%2B24%29\"\"%22%22=%22%22\"\"%288k%5E3%2B12k%5E2%2B4k%29%2B24%28k%5E2%2B2k%2B1%29\"\r\n" );
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document.write( "The first parenthetical expression is what we assumed was a multiple of 24 \r\n" );
document.write( "and the second parenthetical expression is a multiple if 24 and the sum\r\n" );
document.write( "of two multiples of 24 is a multiple of 24.\r\n" );
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document.write( "Thus the theorem is proved by induction.\r\n" );
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document.write( "Edwin
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