document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=672884>Question 672884</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove that for every positive integer n,  n3 + n  is even </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #418378 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=418378\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Case 1: n is an even integer\r<BR>\n" );
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document.write( "Let n be an even integer.\r<BR>\n" );
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document.write( "So n = 2k for some integer k.\r<BR>\n" );
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document.write( "So if n = 2k, then n^3 = (2k)^3 = 8k^3\r<BR>\n" );
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document.write( "and\r<BR>\n" );
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document.write( "n^3 + n\r<BR>\n" );
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document.write( "becomes\r<BR>\n" );
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document.write( "8k^3 + 2k\r<BR>\n" );
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document.write( "which partially factors to\r<BR>\n" );
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document.write( "2(4k^3 + k)\r<BR>\n" );
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document.write( "which is in the form \r<BR>\n" );
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document.write( "2q\r<BR>\n" );
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document.write( "where q = 4k^3 + k (which can be proven that it is also an integer). \r<BR>\n" );
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document.write( "Since 2q is even for any integer q, this proves that if n is an even integer, then n^3+n is even.\r<BR>\n" );
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document.write( "Case 2: n is an odd integer\r<BR>\n" );
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document.write( "Let n be an odd integer.\r<BR>\n" );
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document.write( "So n = 2k+1 for some integer k.\r<BR>\n" );
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document.write( "So if n = 2k+1, then n^3 = (2k+1)^3 = 8k^3 + 12k^2 + 6k + 1\r<BR>\n" );
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document.write( "So\r<BR>\n" );
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document.write( "n^3 + n\r<BR>\n" );
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document.write( "becomes\r<BR>\n" );
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document.write( "(8k^3 + 12k^2 + 6k + 1) + (2k + 1)\r<BR>\n" );
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document.write( "8k^3 + 12k^2 + 6k + 1 + 2k + 1\r<BR>\n" );
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document.write( "8k^3 + 12k^2 + 8k + 2\r<BR>\n" );
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document.write( "which partially factors to\r<BR>\n" );
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document.write( "2(4k^3+6k^2+4k+1)\r<BR>\n" );
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document.write( "which is in the form \r<BR>\n" );
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document.write( "2q\r<BR>\n" );
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document.write( "where q = 4k^3+6k^2+4k+1 (which can be proven that it is also an integer). \r<BR>\n" );
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document.write( "Since 2q is even for any integer q, this proves that if n is an odd integer, then n^3+n is even.\r<BR>\n" );
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document.write( "===========================================================================================================\r<BR>\n" );
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document.write( "We've exhausted all possibilities and scenarios because any integer is either even or odd (cannot be something else or both).\r<BR>\n" );
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document.write( "So these two cases prove that n^3 + n is an even integer for every integer n. </SPAN>\n" );
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