document.write( "Question 478538: Prove that the product of three consecutive integers is divisible by 6; of four consecutive integers by 24. \n" ); document.write( "
Algebra.Com's Answer #327946 by Edwin McCravy(20056)\"\" \"About 
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document.write( "Theorem: If n is a positive integer, then n(n+1)(n+2) is divisible by 3\r\n" );
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document.write( "Proof by induction:\r\n" );
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document.write( "1*2*3 = 6, which is divisible by 6.\r\n" );
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document.write( "Assume that for n = k, k(k+1)(k+2) is divisible by 6\r\n" );
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document.write( "We need to show that, based on that assumption, (k+1)(k+2)(k+3) is also\r\n" );
document.write( "divisible by 6.\r\n" );
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document.write( "(k+1)(k+2)(k+3) = (k+1)(k+2)k + (k+1)(k+2)3 = k(k+1)(k+2) + 3(k+1)(k+2).\r\n" );
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document.write( "By induction hypothesis, the first term is divisible by 6, \r\n" );
document.write( "and the second term 3(k+1)(k+2) is divisible by 6 because it contains\r\n" );
document.write( "a factor 3 and one of the two consecutive integers k+1 or k+2 is\r\n" );
document.write( "even and thus is divisible by 2.  Thus it is divisible by both 3 and\r\n" );
document.write( "2, which means it is divisible by 6.  The theorem is proved since\r\n" );
document.write( "the sum of two multiples of 6 is also a multiple of 6. \r\n" );
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document.write( "Theorem: If n is a positive integer, then n(n+1)(n+2)(n+3) is divisible \r\n" );
document.write( "by 24.\r\n" );
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document.write( "Proof by induction:\r\n" );
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document.write( "1*2*3*4 = 24, which is divisible by 24.\r\n" );
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document.write( "Assume that for n = k, k(k+1)(k+2)(k+3) is divisible by 24\r\n" );
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document.write( "We need to show that (k+1)(k+2)(k+3)(k+4), based on that ssumption,\r\n" );
document.write( "is also divisible by 24.\r\n" );
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document.write( "(k+1)(k+2)(k+3)(k+4) = (k+1)(k+2)(k+3)k + (k+1)(k+2)(k+3)4 = k(k+1)(k+2)(k+3) + 4(k+1)(k+2)(k+3).\r\n" );
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document.write( "By induction hypothesis, the first term is divisible by 24, \r\n" );
document.write( "and the second term 4(k+1)(k+2)(k+3) is divisible by 24 because it contains\r\n" );
document.write( "a factor 4 and by the preceding theorem (k+1)(k+2)(k+3) is divisible by\r\n" );
document.write( "6, so 4(k+2)(k+3)(k+4) is divisible by 24.  Therefore (k+1)(k+2)(k+3)(k+4)\r\n" );
document.write( "is divisible by 24 since it is the sum of two multiples of 24.\r\n" );
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document.write( "Edwin
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