document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=115030>Question 115030</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am struggle with these problems and I need help and explanation on how to do it in the future.\r<BR>\n" );
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document.write( "a) Write the prime factorization of 260.<BR>\n" );
document.write( "b) Find the greatest common factor (GCF) of 84 and 140.<BR>\n" );
document.write( "c) Find the least common multiple (LCM) of 18, 20, 30.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #83685 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=83685\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To do a prime factorization start dividing by the primes in ascending order.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=260%2F2=130 ALIGN=MIDDLE  ALT=\"260%2F2=130\"   DISABLED_style= \"max-width:90%; height:auto;\" >, 1 factor of 2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=130%2F2=65 ALIGN=MIDDLE  ALT=\"130%2F2=65\"   DISABLED_style= \"max-width:90%; height:auto;\" >, 2 factors of 2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=65%2F2 ALIGN=MIDDLE  ALT=\"65%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, not an integer, so no more factors of 2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=65%2F3 ALIGN=MIDDLE  ALT=\"65%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, not an integer, so no factors of 3<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=65%2F5=13 ALIGN=MIDDLE  ALT=\"65%2F5=13\"   DISABLED_style= \"max-width:90%; height:auto;\" >, 1 factor of 5, and the result is prime so you are done.\r<BR>\n" );
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document.write( "The prime factorization is then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=260+=+2%2A2%2A5%2A13 ALIGN=MIDDLE  ALT=\"260+=+2%2A2%2A5%2A13\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In general, when you are performing this process and you don't find any factors, you can quit trying when you are using the largest prime that is less than the square root of the number you are testing.\r<BR>\n" );
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document.write( "The greatest common factor of two numbers is the largest number that divides both of the two numbers evenly.  To find a GCF, perform a prime factorization of each of the original numbers.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=84%2F2=42 ALIGN=MIDDLE  ALT=\"84%2F2=42\"   DISABLED_style= \"max-width:90%; height:auto;\" >, 2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=42%2F2=21 ALIGN=MIDDLE  ALT=\"42%2F2=21\"   DISABLED_style= \"max-width:90%; height:auto;\" >, 2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21%2F2 ALIGN=MIDDLE  ALT=\"21%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, not an integer, so no more factors of 2  (I usually don't write out this step, because I can see the result is odd and will not be divisible by 2 any further)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21%2F3=7 ALIGN=MIDDLE  ALT=\"21%2F3=7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, hence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=84+=+2%2A2%2A3%2A7 ALIGN=MIDDLE  ALT=\"84+=+2%2A2%2A3%2A7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=140%2F2=70 ALIGN=MIDDLE  ALT=\"140%2F2=70\"   DISABLED_style= \"max-width:90%; height:auto;\" >, 2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=70%2F2=35 ALIGN=MIDDLE  ALT=\"70%2F2=35\"   DISABLED_style= \"max-width:90%; height:auto;\" >, 2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=35%2F3 ALIGN=MIDDLE  ALT=\"35%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, not an integer<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=35%2F5=7 ALIGN=MIDDLE  ALT=\"35%2F5=7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, hence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=140=2%2A2%2A5%2A7 ALIGN=MIDDLE  ALT=\"140=2%2A2%2A5%2A7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now examine the prime factors.  The prime factors 84 and 140 have in common are 2, 2, and 7.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A2%2A7=28 ALIGN=MIDDLE  ALT=\"2%2A2%2A7=28\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that is your GCF.\r<BR>\n" );
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document.write( "For the least common multiple, you are looking for the smallest number that is evenly divisible by your numbers.  If you just multiplied 18 * 20 * 30, you would get a common multiple, but it wouldn't necessarily be the smallest.  Again, we fall back on the prime factorization process.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%2F2=9 ALIGN=MIDDLE  ALT=\"18%2F2=9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2F3=3 ALIGN=MIDDLE  ALT=\"9%2F3=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18=2%2A3%2A3 ALIGN=MIDDLE  ALT=\"18=2%2A3%2A3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20%2F2=10 ALIGN=MIDDLE  ALT=\"20%2F2=10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2F2=5 ALIGN=MIDDLE  ALT=\"10%2F2=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20=2%2A2%2A5 ALIGN=MIDDLE  ALT=\"20=2%2A2%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30%2F2=15 ALIGN=MIDDLE  ALT=\"30%2F2=15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%2F3=5 ALIGN=MIDDLE  ALT=\"15%2F3=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30=2%2A3%2A5 ALIGN=MIDDLE  ALT=\"30=2%2A3%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Again, we examine the sets of factors, but this time we need to account for the most instances of a factor.  In the factorization of 18, there are two 3s and at most only one 3 in any of the others, so our LCM will need two 3s, in the factorization of 20, there are two 2s and at most only one 2 in any of the others, so our LCM will also need two 2s, and we have at most one 5 in any of the factorizations, so the LCM will need one 5.  Since there are no other prime factors represented in any of the prime factorizations, we can now say that the LCM is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A2%2A3%2A3%2A5=180 ALIGN=MIDDLE  ALT=\"2%2A2%2A3%2A3%2A5=180\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  You can check that 180 is truly a common multiple by noting that 180 is evenly divisible by 18, 20, and 30.  You may notice that this is considerably (60 times) smaller than just multiplying the three numbers together <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%2A20%2A30=10800 ALIGN=MIDDLE  ALT=\"18%2A20%2A30=10800\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Hope this helps,<BR>\n" );
document.write( "John\r<BR>\n" );
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