document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=97923>Question 97923</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0471321486/algebrahomeworkh>Intro to Real Analysis</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Prove that n^3 + (n+1)^3 + (n+2)^3 is divisible by 9 for all n in Natural numbers.  I have tried, let mathematical induction, let P(n) be the statement, then then n^9 - n is divisible by 9.  I'm stuck...\r<BR>\n" );
document.write( "\n" );
document.write( "Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #71231 by </B><A HREF=/tutors/aboutme.mpl?userid=aaaaaaaa><B>aaaaaaaa(138)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=aaaaaaaa><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=71231\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \"x mod y\" is short for \"remainder of the division of x by y\". Iff \"x mod y\" is 0, x is divisible by y. In our case, y = 9.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E3+%2B+%28n%2B1%29%5E3+%2B+%28n%2B2%29%5E3 ALIGN=MIDDLE  ALT=\"n%5E3+%2B+%28n%2B1%29%5E3+%2B+%28n%2B2%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 9) =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E3+%2B+n%5E3+%2B+3n%5E2+%2B+3n+%2B+1+%2B+n%5E3+%2B+6n%5E2+%2B+12n+%2B+8 ALIGN=MIDDLE  ALT=\"n%5E3+%2B+n%5E3+%2B+3n%5E2+%2B+3n+%2B+1+%2B+n%5E3+%2B+6n%5E2+%2B+12n+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 9)= <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3n%5E3+%2B+9n%5E2+%2B+15n+%2B+9 ALIGN=MIDDLE  ALT=\"3n%5E3+%2B+9n%5E2+%2B+15n+%2B+9\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 9) =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3n%5E3+%2B+9n%5E2+%2B+9n+%2B+6n+%2B+9 ALIGN=MIDDLE  ALT=\"3n%5E3+%2B+9n%5E2+%2B+9n+%2B+6n+%2B+9\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 9)\r<BR>\n" );
document.write( "\n" );
document.write( "9n^2 is always divisible by 9, so we shouldn't worry about that term. The same goes for the \"+9\" at the end, and the \"+9n\" at the middle.\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3n%5E3+%2B+cross%289n%5E2%29+%2B+cross%289n%29+%2B+6n+%2B+cross%289%29 ALIGN=MIDDLE  ALT=\"3n%5E3+%2B+cross%289n%5E2%29+%2B+cross%289n%29+%2B+6n+%2B+cross%289%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 9) =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3n%5E3+%2B+6n ALIGN=MIDDLE  ALT=\"3n%5E3+%2B+6n\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 9)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Therefore,<BR>\n" );
document.write( "P(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E3+%2B+%28n%2B1%29%5E3+%2B+%28n%2B2%29%5E3 ALIGN=MIDDLE  ALT=\"n%5E3+%2B+%28n%2B1%29%5E3+%2B+%28n%2B2%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 9) = 0<BR>\n" );
document.write( "is equivalent to<BR>\n" );
document.write( "S(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3n%5E3+%2B+6n ALIGN=MIDDLE  ALT=\"3n%5E3+%2B+6n\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 9) = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now suppose S(n) is true. I'll try to prove that S(n+1) is also true.\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28n%2B1%29%5E3+%2B+6%28n%2B1%29 ALIGN=MIDDLE  ALT=\"3%28n%2B1%29%5E3+%2B+6%28n%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 9) = <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28n%5E3+%2B+3n%5E2+%2B+3n+%2B+1%29+%2B+6n+%2B+15 ALIGN=MIDDLE  ALT=\"3%28n%5E3+%2B+3n%5E2+%2B+3n+%2B+1%29+%2B+6n+%2B+15\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 9) = <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3n%5E3+%2B+cross%289n%5E2%29+%2B+cross%289n%29+%2B+3+%2B+6n+%2B+15 ALIGN=MIDDLE  ALT=\"3n%5E3+%2B+cross%289n%5E2%29+%2B+cross%289n%29+%2B+3+%2B+6n+%2B+15\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 9) = <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3n%5E3+%2B+6n+%2B+cross%2818%29 ALIGN=MIDDLE  ALT=\"3n%5E3+%2B+6n+%2B+cross%2818%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 9) =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3n%5E3+%2B+6n ALIGN=MIDDLE  ALT=\"3n%5E3+%2B+6n\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 9)\r<BR>\n" );
document.write( "\n" );
document.write( "But wait, if S(n) is true, then\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3n%5E3+%2B+6n ALIGN=MIDDLE  ALT=\"3n%5E3+%2B+6n\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 9) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 9)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The inductive step is done. Now we need to check the base case of natural numbers (which is 0)\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%280%5E3%29+%2B+3%2A0 ALIGN=MIDDLE  ALT=\"3%280%5E3%29+%2B+3%2A0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 9) = <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 9)\r<BR>\n" );
document.write( "\n" );
document.write( "S(x) is therefore true for all naturals.\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );