document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=793660>Question 793660</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you find the product of primes for the number 10080? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #480417 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=480417\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To find the prime factorization for any number you look for prime factors.<BR>\n" );
document.write( "First, is the number divisible by 2?<BR>\n" );
document.write( "If so, divide it by 2, and keep on working with the result.<BR>\n" );
document.write( "If not, try the same with 3; then with 5, 7, 11, 13. 17, 19, 23, and all the other prime numbers, until you get to one whose square exceeds the number.<BR>\n" );
document.write( " <BR>\n" );
document.write( "10080 divides by 2 evenly and the result is 5040, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10080=2%2A5040 ALIGN=MIDDLE  ALT=\"10080=2%2A5040\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "5040 divides by 2 evenly and the result is 2520, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10080=2%2A5040=2%2A2%2A2520 ALIGN=MIDDLE  ALT=\"10080=2%2A5040=2%2A2%2A2520\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2520 divides by 2 evenly and the result is 1260, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10080=2%2A5040=2%2A2%2A2520=2%2A2%2A2%2A1260 ALIGN=MIDDLE  ALT=\"10080=2%2A5040=2%2A2%2A2520=2%2A2%2A2%2A1260\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "1260 divides by 2 evenly and the result is 630, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10080=2%2A5040=2%2A2%2A2520=2%2A2%2A2%2A1260=2%2A2%2A2%2A2%2A630 ALIGN=MIDDLE  ALT=\"10080=2%2A5040=2%2A2%2A2520=2%2A2%2A2%2A1260=2%2A2%2A2%2A2%2A630\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We really do not have to keep writing the products, we just need to keep track of how many times we divide by 2.<BR>\n" );
document.write( "630 divides by 2 evenly and the result is 315.<BR>\n" );
document.write( "315 is not divisible by 2, but divides evenly by 3 and the result is 105.<BR>\n" );
document.write( "105 divides evenly by 3 and the result is 35.<BR>\n" );
document.write( "35 is not divisible by 3, but divides evenly by 5 and the result is 7.<BR>\n" );
document.write( "So, all the factors that we peeled from 10080, like layers from an onion, are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10080=2%2A2%2A2%2A2%2A2%2A3%2A3%2A5%2A7=2%5E5%2A3%5E3%2A5%2A7 ALIGN=MIDDLE  ALT=\"10080=2%2A2%2A2%2A2%2A2%2A3%2A3%2A5%2A7=2%5E5%2A3%5E3%2A5%2A7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To check we can calculate back the product:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E5=32 ALIGN=MIDDLE  ALT=\"2%5E5=32\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E2=9 ALIGN=MIDDLE  ALT=\"3%5E2=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32%2A9%2A5%2A7=10080 ALIGN=MIDDLE  ALT=\"32%2A9%2A5%2A7=10080\"   DISABLED_style= \"max-width:90%; height:auto;\" > according to my calculator.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Of course, if no one asks me to show my work, I take shortcuts, peeling larger numbers in easy chunks. For example I would divide by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10=2%2A5 ALIGN=MIDDLE  ALT=\"10=2%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" > in my head, to get 1008, and would look for factors of 1008, while keeping track of the 2 and 5 (I would write them on scrap paper).<BR>\n" );
document.write( "I may realize that 1008 divides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9=3%2A3 ALIGN=MIDDLE  ALT=\"9=3%2A3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and dividing by 9 would get 112, keeping track of the two 3's in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9=3%2A3 ALIGN=MIDDLE  ALT=\"9=3%2A3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Then I would work with 112 to find the remaining factors. Dividing by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4=2%2A2 ALIGN=MIDDLE  ALT=\"4=2%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" > I would get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=28=4%2A7=2%2A2%2A7 ALIGN=MIDDLE  ALT=\"28=4%2A7=2%2A2%2A7\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So the factor found were 2 and 5 first (as factors of 10),<BR>\n" );
document.write( "3 and 3 next (in 9),<BR>\n" );
document.write( "2 and 2 next (in 4),<BR>\n" );
document.write( "and finally 2, 2, and 7 (in 28).<BR>\n" );
document.write( "In all, 2 appeared 5 times, 3 appeared twice, 5 once, and 7 once,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10080=2%5E5%2A3%5E2%2A5%2A7 ALIGN=MIDDLE  ALT=\"10080=2%5E5%2A3%5E2%2A5%2A7\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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