document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=254068>Question 254068</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that only one of the numbers n, n+2, n+4 is divisible by 3. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #186396 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=186396\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "You didn't say, but the only way this works is if <i>n</i> is an integer.  Use the modulo function. <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n \text{ mod } m\"> returns the remainder when <i>n</i> is divided by <i>m</i>.  For positive integers, the range of the modulo function is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \{0, 1, 2, \cdots, m\,-\,1\}\">, therefore for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n \text{ mod } 3\">, the range is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \{0, 1, 2\}\">.\r<BR>\n" );
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document.write( "So there are three possibilities regardless of the value of <i>n</i>:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ n \text{ mod } 3\ =\ 0\ \Rightarrow\ n\,+\,2 \text{ mod } 3\ =\ 2 \text{ and } n\,+\,4 \text{ mod } 3\ =\ 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ n \text{ mod } 3\ =\ 1\ \Rightarrow\ n\,+\,2 \text{ mod } 3\ =\ 0 \text{ and } n\,+\,4 \text{ mod } 3\ =\ 2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ n \text{ mod } 3\ =\ 2\ \Rightarrow\ n\,+\,2 \text{ mod } 3\ =\ 1 \text{ and } n\,+\,4 \text{ mod } 3\ =\ 0\">\r<BR>\n" );
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document.write( "For each of the three possibilities, there is only one 0 result, hence only one of the numbers is divisible by 3.\r<BR>\n" );
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document.write( "If <i>n</i> is a negative integer, the range of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n \text{ mod } 3\"> is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \{-2, -1, 0\}\"> and the result is the same.\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" );
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