document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1126866>Question 1126866</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The product of all the ages of the teenagers at a party was 2 971 987 200. The number of 18 year olds who attended the party? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #743196 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=743196\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Cool problem!<br><BR>\n" );
document.write( "I think the other tutor completely missed the point of the problem....<br><BR>\n" );
document.write( "We need to find the prime factorization of the number and then put the prime factors together so that all the ages are in the range 13-19.<br><BR>\n" );
document.write( "The prime factorization is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2971987200+=+%282%5E8%29%283%5E6%29%285%5E2%29%287%5E2%29%2813%29 ALIGN=MIDDLE  ALT=\"2971987200+=+%282%5E8%29%283%5E6%29%285%5E2%29%287%5E2%29%2813%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(1) Clearly there must be exactly one 13-year-old.<br><BR>\n" );
document.write( "The remaining prime factors are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%5E8%29%283%5E6%29%285%5E2%29%287%5E2%29 ALIGN=MIDDLE  ALT=\"%282%5E8%29%283%5E6%29%285%5E2%29%287%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(2) The only multiple of 7 in the required range is 14; since there are 2 factors of 7, there must be 2 14-year-olds.  That uses both factors of 7 and 2 of the factors of 2.<br><BR>\n" );
document.write( "The remaining prime factors are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%5E6%29%283%5E6%29%285%5E2%29 ALIGN=MIDDLE  ALT=\"%282%5E6%29%283%5E6%29%285%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(3) The only multiple of 5 in the required range is 15; since there are 2 factors of 5, there must be 2 15-year olds.  That uses both factors of 5 and 2 of the factors of 3.<br><BR>\n" );
document.write( "The remaining prime factors are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%5E6%29%283%5E4%29 ALIGN=MIDDLE  ALT=\"%282%5E6%29%283%5E4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(4) The only age in the required range that is a multiple of 3 and has only 2 and 3 as prime factors is 18; that age requires 1 factor of 2 and 2 factors of 3.  Since there are 4 factors of 3 remaining, there must be 2 18-year-olds.  That uses all 4 factors of 3 and 2 of the remaining 6 factors of 2.<br><BR>\n" );
document.write( "The remaining prime factors are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E4=16 ALIGN=MIDDLE  ALT=\"2%5E4=16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So there is 1 16-year-old.<br><BR>\n" );
document.write( "SOLVED!<br><BR>\n" );
document.write( "The ages of the teenagers at the party:<BR>\n" );
document.write( "13: 1<BR>\n" );
document.write( "14: 2<BR>\n" );
document.write( "15: 2<BR>\n" );
document.write( "16: 1<BR>\n" );
document.write( "18: 2<br><BR>\n" );
document.write( "CHECK: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2813%29%2814%5E2%29%2815%5E2%29%2816%29%2818%5E2%29+=+2871987200 ALIGN=MIDDLE  ALT=\"%2813%29%2814%5E2%29%2815%5E2%29%2816%29%2818%5E2%29+=+2871987200\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Answer to the question that was asked: there are two 18-year-olds at the party. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );