document.write( "Question 242615: show that only one out of n, n+2 or n+4 is divisible by 3 where n is positive integer. \n" ); document.write( "
Algebra.Com's Answer #177882 by Edwin McCravy(20056)\"\" \"About 
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show that only one out of n, n+2 or n+4 is divisible by 3 where n is positive integer.\r
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document.write( "By induction:\r\n" );
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document.write( "When n = 1, exactly one of 1, 1+2, and 1+4 is divisible by 3,\r\n" );
document.write( "namely 1+2, since 3 is divisible by 3, and the other two 1 and\r\n" );
document.write( "5 are not.\r\n" );
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document.write( "Suppose for n < k, only one out of n, n+2, n+4 is divisible by 3\r\n" );
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document.write( "For n = k, we consider k, k+2, k+4.  \r\n" );
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document.write( "By the induction hypothesis, only one of k-1, k+1 and k+3 is \r\n" );
document.write( "divisible by 3.  We look at the three possible cases.\r\n" );
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document.write( "Case 1: k-1 is the one which is divisible by 3. Then k-1 = 3m, for\r\n" );
document.write( "some positive integer m.  \r\n" );
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document.write( "Then add 1 to both sides of k-1 = 3m:\r\n" );
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document.write( "k-1+1=3m+1\r\n" );
document.write( "k = 3m+1\r\n" );
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document.write( "then k is NOT divisible by 3\r\n" );
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document.write( "Now add 3 to both sides of k-1 = 3m:\r\n" );
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document.write( "k-1+3=3m+3\r\n" );
document.write( "k+2 = 3m+3\r\n" );
document.write( "k+2 = 3(m+1)\r\n" );
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document.write( "then k+2 IS divisible by 3\r\n" );
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document.write( "Now add 5 to both sides of k-1 = 3m:\r\n" );
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document.write( "k-1+5=3m+5\r\n" );
document.write( "k+4 = 3m+5\r\n" );
document.write( "k+4 = 3m+3 + 2 = 3(m+1)+2\r\n" );
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document.write( "then k+4 is NOT divisible by 3.\r\n" );
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document.write( "So, we have proved case 1 for n = k\r\n" );
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document.write( "Case 2: k+1 is the one which is divisible by 3. Then k+1 = 3m, for\r\n" );
document.write( "some positive integer m.  \r\n" );
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document.write( "Then add -1 to both sides of k+1 = 3m:\r\n" );
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document.write( "k+1-1=3m-1\r\n" );
document.write( "k = 3m-1\r\n" );
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document.write( "then k is NOT divisible by 3\r\n" );
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document.write( "Now add 1 to both sides of k+1 = 3m:\r\n" );
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document.write( "k+1+1=3m+1\r\n" );
document.write( "k+2 = 3m+1\r\n" );
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document.write( "then k+2 is NOT divisible by 3\r\n" );
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document.write( "Now add 3 to both sides of k+1 = 3m:\r\n" );
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document.write( "k+1+3=3m+3\r\n" );
document.write( "k+4 = 3m+3\r\n" );
document.write( "k+4 = 3(m+1)\r\n" );
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document.write( "so k+4 IS divisible by 3.\r\n" );
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document.write( "So, we have proved case 2.\r\n" );
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document.write( "Case 3: k+3 is the one which is divisible by 3. Then k+3 = 3m, for\r\n" );
document.write( "some positive integer m.  \r\n" );
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document.write( "Then add -3 to both sides of k+3 = 3m:\r\n" );
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document.write( "k+3-3=3m-3\r\n" );
document.write( "k = 3(m-1)\r\n" );
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document.write( "then k IS divisible by 3\r\n" );
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document.write( "Now add -1 to both sides of k+3 = 3m:\r\n" );
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document.write( "k+3-1=3m-1\r\n" );
document.write( "k+2 = 3m-1\r\n" );
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document.write( "then k+2 is NOT divisible by 3\r\n" );
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document.write( "Now add 1 to both sides of k+3 = 3m:\r\n" );
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document.write( "k+3+1=3m+1\r\n" );
document.write( "k+4 = 3m+1\r\n" );
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document.write( "then k+4 is NOT divisible by 3.\r\n" );
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document.write( "So, we have proved case 3.\r\n" );
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document.write( "The proof is now complete.\r\n" );
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document.write( "Edwin
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