document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=884963>Question 884963</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the sum of all positive integers which are less than 2010 and are not 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 #534820 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=534820\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First, I looked at the numbers starting with 3<BR>\n" );
document.write( "that ARE divisible by 3:<BR>\n" );
document.write( "3, 6, 9, 12, 15, . . .<BR>\n" );
document.write( "----------------------<BR>\n" );
document.write( "I noticed that:<BR>\n" );
document.write( "3 = 1 + 2<BR>\n" );
document.write( "6 = 2 + 4 <BR>\n" );
document.write( "9 = 3 + 6 <BR>\n" );
document.write( "12 = 4 + 8<BR>\n" );
document.write( " . . .<BR>\n" );
document.write( "-------------<BR>\n" );
document.write( "And then:<BR>\n" );
document.write( "3 = 1 + 1 + 1<BR>\n" );
document.write( "6 = 2 + 2 + 2<BR>\n" );
document.write( "9 = 3 + 3 + 3<BR>\n" );
document.write( "12 = 4 + 4 + 4<BR>\n" );
document.write( " . . .<BR>\n" );
document.write( "----------------<BR>\n" );
document.write( "How MANY numbers less than 2010<BR>\n" );
document.write( "are divisible by 3?<BR>\n" );
document.write( "That would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2010+%2F+3+=+670+-+1+=+669+ ALIGN=MIDDLE  ALT=\"+2010+%2F+3+=+670+-+1+=+669+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "------------------------------------------<BR>\n" );
document.write( "I know how to find the sum 1 + 2 + 3 + 4 + 5 + . . . + n<BR>\n" );
document.write( "It is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28+n%2A%28+n+%2B+1+%29+%29+%2F+2+ ALIGN=MIDDLE  ALT=\"+%28+n%2A%28+n+%2B+1+%29+%29+%2F+2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In the case of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+n+=+5+ ALIGN=MIDDLE  ALT=\"+n+=+5+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, it is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28+5%2A6+%29+%2F+2+=+15+ ALIGN=MIDDLE  ALT=\"+%28+5%2A6+%29+%2F+2+=+15+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "---------------------------<BR>\n" );
document.write( "So, it looks like I need to find<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28+n%5B1%5D%2A%28+n%5B1%5D+%2B+1+%29+%29+%2F+2+ ALIGN=MIDDLE  ALT=\"+%28+n%5B1%5D%2A%28+n%5B1%5D+%2B+1+%29+%29+%2F+2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+n%5B1%5D+=+2009+ ALIGN=MIDDLE  ALT=\"+n%5B1%5D+=+2009+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then I need to subtract:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3%2A%28%28+n%5B2%5D%2A%28+n%5B2%5D+%2B+1+%29+%29+%2F+2%29+ ALIGN=MIDDLE  ALT=\"+3%2A%28%28+n%5B2%5D%2A%28+n%5B2%5D+%2B+1+%29+%29+%2F+2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+n%5B2%5D+=+669+ ALIGN=MIDDLE  ALT=\"+n%5B2%5D+=+669+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "----------------------------------------------<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28+2009%2A2010+%29+%2F+2+-+%283%2F2%29%2A%28+669%2A610+%29+ ALIGN=MIDDLE  ALT=\"+%28+2009%2A2010+%29+%2F+2+-+%283%2F2%29%2A%28+669%2A610+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+4038090+%2F+2+-+%28+3%2F2+%29%2A408090+ ALIGN=MIDDLE  ALT=\"+4038090+%2F+2+-+%28+3%2F2+%29%2A408090+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2019045+-+612135+=+1406910+ ALIGN=MIDDLE  ALT=\"+2019045+-+612135+=+1406910+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I get 1,406,910 for the sum<BR>\n" );
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document.write( "Maybe there is an easier way to do this, though<BR>\n" );
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document.write( "Checking it for less than 1 through 20, you have<BR>\n" );
document.write( "1 + 2 + 4 + 5 + 7 + 8 + 10 + 11 + 13 + 14 + 16 + 17 + 19<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+127+ ALIGN=MIDDLE  ALT=\"+127+\"   DISABLED_style= \"max-width:90%; height:auto;\" > total<BR>\n" );
document.write( "----------------<BR>\n" );
document.write( "Using the formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+19%2F3+=+6+ ALIGN=MIDDLE  ALT=\"+19%2F3+=+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( nearest whole number )<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28+n%5B1%5D%2A%28+n%5B1%5D+%2B+1+%29+%29+%2F+2+ ALIGN=MIDDLE  ALT=\"+%28+n%5B1%5D%2A%28+n%5B1%5D+%2B+1+%29+%29+%2F+2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28+19%2A20+%29+%2F+2+=+190+ ALIGN=MIDDLE  ALT=\"+%28+19%2A20+%29+%2F+2+=+190+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3%2A%28%28+n%5B2%5D%2A%28+n%5B2%5D+%2B+1+%29+%29+%2F+2%29+ ALIGN=MIDDLE  ALT=\"+3%2A%28%28+n%5B2%5D%2A%28+n%5B2%5D+%2B+1+%29+%29+%2F+2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28+3%2F2%29%2A%28+6%2A7+%29+=+63+ ALIGN=MIDDLE  ALT=\"+%28+3%2F2%29%2A%28+6%2A7+%29+=+63+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+190+-+63+=+127+ ALIGN=MIDDLE  ALT=\"+190+-+63+=+127+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "seems to work\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
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document.write( " </SPAN>\n" );
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