document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=800903>Question 800903</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A circle has 29 points arranged in a clockwise manner numbered from 0 to 28, as shown<BR>\n" );
document.write( "in the figure below. A bug moves clockwise around the circle according to the following<BR>\n" );
document.write( "rule. If it is at a point i on the circle, it moves clockwise in 1 second by ( 1 + r ) places,<BR>\n" );
document.write( "where r is the reminder ( possibly 0 ) when i is divided by 11. Thus if it is at position 5, it<BR>\n" );
document.write( "moves clockwise in one second by ( 1 + 5 ) places to point 11. Similarly if it is at position<BR>\n" );
document.write( "28 it moves ( 1 + 6 ) or 7 places to point 6 in one second.<BR>\n" );
document.write( "If it starts at point 23, at what point will it be after 2012 seconds?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #483368 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=483368\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If the bug starts at point 23,<BR>\n" );
document.write( "since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=23=22%2B1=2%2A11%2B1 ALIGN=MIDDLE  ALT=\"23=22%2B1=2%2A11%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "the bug will move <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B1=2 ALIGN=MIDDLE  ALT=\"1%2B1=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > places at 1 second,<BR>\n" );
document.write( "and will end at point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=23%2B2=25 ALIGN=MIDDLE  ALT=\"23%2B2=25\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25=22%2B3 ALIGN=MIDDLE  ALT=\"25=22%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "the bug will move <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B1=4 ALIGN=MIDDLE  ALT=\"3%2B1=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > places at 2 seconds,<BR>\n" );
document.write( "and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%2B4=29 ALIGN=MIDDLE  ALT=\"25%2B4=29\"   DISABLED_style= \"max-width:90%; height:auto;\" > it will end at point 0.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=0%2A11%2B0 ALIGN=MIDDLE  ALT=\"0=0%2A11%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "the bug will move <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2B1=1 ALIGN=MIDDLE  ALT=\"0%2B1=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > place at 3 seconds,<BR>\n" );
document.write( "and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2B1=1 ALIGN=MIDDLE  ALT=\"0%2B1=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > it will end at point 1.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1=0%2B1 ALIGN=MIDDLE  ALT=\"1=0%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "the bug will move <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B1=2 ALIGN=MIDDLE  ALT=\"1%2B1=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > places at 4 seconds,<BR>\n" );
document.write( "and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B2=3 ALIGN=MIDDLE  ALT=\"1%2B2=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > it will end at point 3.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3=0%2B3 ALIGN=MIDDLE  ALT=\"3=0%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "the bug will move <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B1=4 ALIGN=MIDDLE  ALT=\"3%2B1=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > places at 5 seconds,<BR>\n" );
document.write( "and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B4=7 ALIGN=MIDDLE  ALT=\"3%2B4=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > it will end at point 7.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7=0%2B7 ALIGN=MIDDLE  ALT=\"7=0%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "the bug will move <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2B1=8 ALIGN=MIDDLE  ALT=\"7%2B1=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > places at 6 seconds,<BR>\n" );
document.write( "and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2B8=15 ALIGN=MIDDLE  ALT=\"7%2B8=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > it will end at point 15.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15=11%2B4=1%2A11%2B4 ALIGN=MIDDLE  ALT=\"15=11%2B4=1%2A11%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "the bug will move <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B1=5 ALIGN=MIDDLE  ALT=\"4%2B1=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > places at 7 seconds,<BR>\n" );
document.write( "and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%2B5=20 ALIGN=MIDDLE  ALT=\"15%2B5=20\"   DISABLED_style= \"max-width:90%; height:auto;\" > it will end at point 20.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20=11%2B9 ALIGN=MIDDLE  ALT=\"20=11%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "the bug will move <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2B1=10 ALIGN=MIDDLE  ALT=\"9%2B1=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > places at 8 seconds,<BR>\n" );
document.write( "and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20%2B10=30=29%2B1 ALIGN=MIDDLE  ALT=\"20%2B10=30=29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > the bug will go past 0 will end at point 1.<BR>\n" );
document.write( " <BR>\n" );
document.write( "From that point on, life will become very boring for the bug (and for anyone watching that bug), because the bug will go through endlessly repeated cycles visiting points 3, 7, 15, 20, and 1, in that order, again, and again, and again.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Each cycle will take 5 seconds, so after landing on point 1 at 3 seconds, the bug will keep cycling through points 3, 7, 15, 20, and 1, returning to point 1 every 5 seconds.<BR>\n" );
document.write( " <BR>\n" );
document.write( "In the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2012-3=2009 ALIGN=MIDDLE  ALT=\"2012-3=2009\"   DISABLED_style= \"max-width:90%; height:auto;\" > seconds after landing on point 1 for the first time,<BR>\n" );
document.write( "the bug will return to point 1 401 more times (2009 divided by 5 gives a quotient of 401),<BR>\n" );
document.write( "the last two times at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B400%2A5=3%2B2000=2003 ALIGN=MIDDLE  ALT=\"3%2B400%2A5=3%2B2000=2003\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B401%2A5=3%2B2005=2008 ALIGN=MIDDLE  ALT=\"3%2B401%2A5=3%2B2005=2008\"   DISABLED_style= \"max-width:90%; height:auto;\" > seconds.<BR>\n" );
document.write( "In 5 more seconds, at 2013 seconds, the bug would return to point 1 once again,<BR>\n" );
document.write( "but at 2012 seconds it had landed on point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2820%29 ALIGN=MIDDLE  ALT=\"highlight%2820%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "I believe that \"after 2012 seconds\" the bug will rest for a while at point 20,<BR>\n" );
document.write( "maybe for half a second, before moving to point 1 again.<BR>\n" );
document.write( "After all if the bug moved continuously, \"after 2012 seconds\" would have no meaning.  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );