document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=731432>Question 731432</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What five different whole numbers add up to 18 multiply those same numbers and get 240? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #447229 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=447229\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> THE ANSWER:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%281%2B2%2B4%2B5%2B6=18%29 ALIGN=MIDDLE  ALT=\"highlight%281%2B2%2B4%2B5%2B6=18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%281%2A2%2A4%2A5%2A6=240%29 ALIGN=MIDDLE  ALT=\"highlight%281%2A2%2A4%2A5%2A6=240%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "HOW TO GET TO THE ANSWER:<BR>\n" );
document.write( "We need to find factors of 240.<BR>\n" );
document.write( "Either systematically or sloppily.<BR>\n" );
document.write( "Systematically:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=240=2%5E4%2A3%2A5 ALIGN=MIDDLE  ALT=\"240=2%5E4%2A3%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" > prime factorization that can be obtained by dividing over and over by prime numbers in order, starting from 2 and going up:<BR>\n" );
document.write( "240 ÷ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%282%29 ALIGN=MIDDLE  ALT=\"red%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 120<BR>\n" );
document.write( "120 ÷ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%282%29 ALIGN=MIDDLE  ALT=\"red%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 60<BR>\n" );
document.write( "60 ÷ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%282%29 ALIGN=MIDDLE  ALT=\"red%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 30<BR>\n" );
document.write( "30 ÷ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%282%29 ALIGN=MIDDLE  ALT=\"red%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 15<BR>\n" );
document.write( "15 ÷ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%283%29 ALIGN=MIDDLE  ALT=\"red%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%285%29 ALIGN=MIDDLE  ALT=\"red%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So 240 = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%282%29 ALIGN=MIDDLE  ALT=\"red%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >X<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%282%29 ALIGN=MIDDLE  ALT=\"red%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >X<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%282%29 ALIGN=MIDDLE  ALT=\"red%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >X<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%282%29 ALIGN=MIDDLE  ALT=\"red%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >X<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%283%29 ALIGN=MIDDLE  ALT=\"red%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >X<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%285%29 ALIGN=MIDDLE  ALT=\"red%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The sloppy way:<BR>\n" );
document.write( "I know that 240 = 30 X 8, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24=8%2A30=8%2A10%2A3=8%2A2%2A5%2A3=2%2A2%2A2%2A2%2A5%2A3 ALIGN=MIDDLE  ALT=\"24=8%2A30=8%2A10%2A3=8%2A2%2A5%2A3=2%2A2%2A2%2A2%2A5%2A3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Then, it's a question of turning the 6 factors in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A2%2A2%2A2%2A3%2A5 ALIGN=MIDDLE  ALT=\"2%2A2%2A2%2A2%2A3%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" > into five factors by grouping (multiplying) some together, and maybe including a 1 as a factor.<BR>\n" );
document.write( "If we do not include a 1, to add up to the even number 18, we could have both odd factors (3 and 5) by themselves, adding to 12, or both paired with a 2, to become even numbers.<BR>\n" );
document.write( "Neither choice works.<BR>\n" );
document.write( "With 3 and 5 as two of the five different whole numbers, we have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B5=8 ALIGN=MIDDLE  ALT=\"3%2B5=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the four remaining 2's cannot be grouped into the three additional different whole numbers.<BR>\n" );
document.write( "Using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A3=6 ALIGN=MIDDLE  ALT=\"2%2A3=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A5=10 ALIGN=MIDDLE  ALT=\"2%2A5=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > as two of the five different whole numbers is not possible either, because we would just have two extra 2's to make three additional whole numbers from.<BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%281%29 ALIGN=MIDDLE  ALT=\"blue%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as one of the five different whole numbers, what would we do with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" >?<BR>\n" );
document.write( "We need one and only one additional odd number.<BR>\n" );
document.write( "The other odd number must be turned into an even number by grouping it with a 2.<BR>\n" );
document.write( "We would be left with three 2's to be turned into the two additional different whole numbers.<BR>\n" );
document.write( "Those two additional different whole numbers can only be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%282%29 ALIGN=MIDDLE  ALT=\"blue%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A2=blue%284%29 ALIGN=MIDDLE  ALT=\"2%2A2=blue%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "What do we do with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" >?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A3=blue%286%29 ALIGN=MIDDLE  ALT=\"2%2A3=blue%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%285%29 ALIGN=MIDDLE  ALT=\"blue%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as two of the five different whole numbers is an obvious choice, because using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A5=10 ALIGN=MIDDLE  ALT=\"2%2A5=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > would make the sum too large:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B3%2B10%2B2%2B4=14%2B2%2B4=20 ALIGN=MIDDLE  ALT=\"1%2B3%2B10%2B2%2B4=14%2B2%2B4=20\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28blue%281%29%29 ALIGN=MIDDLE  ALT=\"highlight%28blue%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28blue%282%29%29 ALIGN=MIDDLE  ALT=\"highlight%28blue%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28blue%284%29%29 ALIGN=MIDDLE  ALT=\"highlight%28blue%284%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28blue%285%29%29 ALIGN=MIDDLE  ALT=\"highlight%28blue%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28blue%286%29%29 ALIGN=MIDDLE  ALT=\"highlight%28blue%286%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are the five different whole numbers that multiply to yield 240,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%281%2A2%2A4%2A5%2A6=240%29 ALIGN=MIDDLE  ALT=\"highlight%281%2A2%2A4%2A5%2A6=240%29\"   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=highlight%281%2B2%2B4%2B5%2B6=18%29 ALIGN=MIDDLE  ALT=\"highlight%281%2B2%2B4%2B5%2B6=18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the 5-number sum. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );