document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=903349>Question 903349</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove: If n is an integer, then n^2 + n^3 is an even number. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #547923 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=547923\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Prove: If n is an integer, then n^2 + n^3 is an even number.\r<BR>\n" );
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document.write( "Case A: n is 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( "n = 2k\r<BR>\n" );
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document.write( "n^2 = (2k)^2\r<BR>\n" );
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document.write( "n^2 = 4k^2\r<BR>\n" );
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document.write( "n = 2k\r<BR>\n" );
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document.write( "n^3 = (2k)^3\r<BR>\n" );
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document.write( "n^3 = 8k^3\r<BR>\n" );
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document.write( "n^2 + n^3 = 4k^2 + 8k^3\r<BR>\n" );
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document.write( "n^2 + n^3 = 2*(2k^2 + 4k^3)\r<BR>\n" );
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document.write( "that proves n^2 + n^3 is even since 2 is a factor.\r<BR>\n" );
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document.write( "Case B: n is an odd integer\r<BR>\n" );
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document.write( "n = 2m+1 for some integer m\r<BR>\n" );
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document.write( "n = 2m+1\r<BR>\n" );
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document.write( "n^2 = (2m+1)^2\r<BR>\n" );
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document.write( "n^2 = 4m^2 + 4m + 1\r<BR>\n" );
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document.write( "n = 2m+1\r<BR>\n" );
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document.write( "n^3 = (2m+1)^3\r<BR>\n" );
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document.write( "n^3 = 8m^3+12m^2+6m+1\r<BR>\n" );
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document.write( "n^2 + n^3 = 4m^2 + 4m + 1 + 8m^3+12m^2+6m+1\r<BR>\n" );
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document.write( "n^2 + n^3 = 8m^3+16m^2+10m+2\r<BR>\n" );
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document.write( "n^2 + n^3 = 2*(4m^3+8m^2+5m+1)\r<BR>\n" );
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document.write( "that proves n^2 + n^3 is even since 2 is a factor.\r<BR>\n" );
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document.write( "That wraps up the proof because there are only two possible cases if n is an integer.<BR>\n" );
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