document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209222>Question 1209222</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a,b and c are positive integers, find the sum a+b+c if:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3b%5E2c+=+3528%2C+ab%5E3c%5E2+=+55566 ALIGN=MIDDLE  ALT=\"a%5E3b%5E2c+=+3528%2C+ab%5E3c%5E2+=+55566\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2bc%5E3+=+20412 ALIGN=MIDDLE  ALT=\"a%5E2bc%5E3+=+20412\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847867 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=847867\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Using either pencil and paper or an online prime factorization tool, find that<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20412=%282%5E2%29%283%5E6%29%287%5E1%29=%28a%5E2%29%28b%29%28c%5E3%29 ALIGN=MIDDLE  ALT=\"20412=%282%5E2%29%283%5E6%29%287%5E1%29=%28a%5E2%29%28b%29%28c%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We don't need to find the prime factorization of the other two numbers; the answer is uniquely determined by this one.<br><BR>\n" );
document.write( "In that prime factorization....<BR>\n" );
document.write( "(1) prime factors 2 and 3 are both used more than once; and factors a and c are both used more than once.  7 and b are the only factors used once each, so b must be 7<BR>\n" );
document.write( "(2) prime factor 2 is used twice and factor c is used 3 times, so c can't be 2<br><BR>\n" );
document.write( "So we must have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E2=a%5E2 ALIGN=MIDDLE  ALT=\"2%5E2=a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > so a = 2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E6=c%5E3 ALIGN=MIDDLE  ALT=\"3%5E6=c%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > so c = 3^2 = 9<br><BR>\n" );
document.write( "ANSWER: (a,b,c) = (2,7,9)<br><BR>\n" );
document.write( "CHECK:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%5E3%29%28b%5E2%29%28c%29=8%2A49%2A9=3528 ALIGN=MIDDLE  ALT=\"%28a%5E3%29%28b%5E2%29%28c%29=8%2A49%2A9=3528\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%29%28b%5E3%29%28c%5E2%29=2%2A343%2A81=55566 ALIGN=MIDDLE  ALT=\"%28a%29%28b%5E3%29%28c%5E2%29=2%2A343%2A81=55566\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%5E2%29%28b%29%28c%5E3%29=4%2A7%2A729=20412 ALIGN=MIDDLE  ALT=\"%28a%5E2%29%28b%29%28c%5E3%29=4%2A7%2A729=20412\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );