document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182800>Question 1182800</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many pairs (a,b) of positive integers are there such that a and b are factors of 6^6 and a is a factor of b? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #812889 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=812889\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
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document.write( "Yes; I misread the problem, reading it to say that the product of a and b is 6^6.<br><BR>\n" );
document.write( "So my response below is not appropriate for the problem as given.<br><BR>\n" );
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document.write( "I have no idea what problem the other tutor was solving; the way I read the problem, her solution is not of the given problem.<br><BR>\n" );
document.write( "We have the number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%5E6+=+%282%5E6%29%283%5E6%29 ALIGN=MIDDLE  ALT=\"6%5E6+=+%282%5E6%29%283%5E6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "a and b are factors of that number, so a and b are numbers of the form<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+%282%5Ep%29%283%5Eq%29 ALIGN=MIDDLE  ALT=\"a+=+%282%5Ep%29%283%5Eq%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+%282%5Er%29%283%5Es%29 ALIGN=MIDDLE  ALT=\"b+=+%282%5Er%29%283%5Es%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "where p+r=6 and q+s=6.<br><BR>\n" );
document.write( "We also need to have a being a factor of b.  That means<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%3C=r ALIGN=MIDDLE  ALT=\"p%3C=r\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q%3C=s ALIGN=MIDDLE  ALT=\"q%3C=s\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The requirements p+r=6 and p<=r give us 4 choices for p -- 0, 1, 2, or 3.<BR>\n" );
document.write( "Likewise the requirements q+s=6 and q<=s give use the same 4 choices for q.<br><BR>\n" );
document.write( "So the total number of pairs (a,b) that satisfy the conditions of the problem is 4*4=16.<br><BR>\n" );
document.write( " </SPAN>\n" );
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