document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=977112>Question 977112</A>: <I><SPAN STYLE=\"word-break: break-all;\"> show that 3 divides  exactly any one of n,n+1 or n+3? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #598730 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=598730\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "I think you meant the claim to be that 3 divides one of n, n + 1, or n + 2 since if 3 divides n it must, perforce divide n + 3.\r<BR>\n" );
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document.write( "Claim:  For all integers n, 3 divides exactly one of n, n + 1, or n + 2\r<BR>\n" );
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document.write( "We prove by exhaustive cases using the Generalized Divisibility Theorem\r<BR>\n" );
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document.write( "<b><i>Generalized Divisibility Theorem</i></b>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \forall\ a,\,b\ \in\ \mathbb{Z},\,b\ \not =\ 0, \exists\ q,\,r\ \in\ \mathbb{Z}\"> such that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a\ =\ q\cdot b\ +\ r\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 0\ \leq r\ <\ |b|\">\r<BR>\n" );
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document.write( "From which it follows that all integers can be represented by one of the forms:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 3b\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 3b\ +\ 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 3b\ +\ 2\">\r<BR>\n" );
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document.write( "for for some integer <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\">\r<BR>\n" );
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document.write( "Cases:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE n\ =\ 3b\ \Right\ 3|n\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE n\ =\ 3b\ +\ 2 \Right\ n\ +\ 1\ =\ 3b\ +\ 3\ \Right\ 3|(n\ +\ 1)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE n\ =\ 3b\ +\ 1 \Right n\ +\ 2\ =\ 3b\ +\ 3\ \Right 3|(n\ +\ 2)\">\r<BR>\n" );
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document.write( "The three cases are exhaustive and each case has the same conclusion, hence the claim is proven.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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