document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=739966>Question 739966</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the sum of all natural numbers between 1 and 100 which are not exactly divisible by 2 or 3? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #451499 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=451499\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We can find that sum if we find the sum for all the natural numbers between i and 100, and for the sums for those numbers that are not supposed to be counted.<BR>\n" );
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document.write( "The sum of all integers from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28i%2Ci=1%2Ci=n%29=n%28n%2B1%29%2F2 ALIGN=MIDDLE  ALT=\"sum%28i%2Ci=1%2Ci=n%29=n%28n%2B1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The sum of all natural numbers between 1 and 100 is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100%2A101%2F2=50%2A101=5050 ALIGN=MIDDLE  ALT=\"100%2A101%2F2=50%2A101=5050\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The sum of all natural numbers between 1 and 100 that are exactly divisible by 2 is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%282i%2Ci=1%2Ci=50%29=2%2Asum%28i%2Ci=1%2Ci=50%29=2%2A%2850%2A51%2F2%29=50%2A51=2550 ALIGN=MIDDLE  ALT=\"sum%282i%2Ci=1%2Ci=50%29=2%2Asum%28i%2Ci=1%2Ci=50%29=2%2A%2850%2A51%2F2%29=50%2A51=2550\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The sum of all natural numbers between 1 and 100 that are exactly divisible by 3 is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%283i%2Ci=1%2Ci=33%29=3%2Asum%28i%2Ci=1%2Ci=33%29=3%2A%2833%2A34%2F2%29=99%2A%2834%2F2%29=99%2A17=1683 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "All numbers that are exactly divisible by 2 and 3 are divisible by 6, and all numbers exactly divisible by 6 are divisible by 2 and 3.<BR>\n" );
document.write( "The sum of all natural numbers between 1 and 100 that are exactly divisible by 2 and 3 is the sum of all natural numbers between 1 and 100 that are exactly divisible by 6, which can be calculated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%286i%2Ci=2%2Ci=16%29=6%2Asum%28i%2Ci=1%2Ci=16%29=6%2A%2816%2A17%2F2%29=48%2A17=816 ALIGN=MIDDLE  ALT=\"sum%286i%2Ci=2%2Ci=16%29=6%2Asum%28i%2Ci=1%2Ci=16%29=6%2A%2816%2A17%2F2%29=48%2A17=816\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "To find the sum we want, we start with the sum of all the natural numbers from 1 to 100;<BR>\n" );
document.write( "we subtract the sum of those that are exactly divisible by 2;<BR>\n" );
document.write( "we subtract the sum of those that are exactly divisible by 3,<BR>\n" );
document.write( "and since in the previous subtractions we subtracted the numbers divisible bt 6 twice,<BR>\n" );
document.write( "we add the sum of natural numbers between 1 and 100 that are exactly divisible by 6.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50%2A101-50%2A51-99%2A17%2B48%2A17=50%2A%28101-51%29-%2899-48%2917=50%2A50-51%2A17=2500-867=highlight%281633%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or if you prefer, you can calculate it as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5050-2550-1683%2B816=2500-99%2A17%2B48%2A17=highlight%281633%29 ALIGN=MIDDLE  ALT=\"5050-2550-1683%2B816=2500-99%2A17%2B48%2A17=highlight%281633%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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