document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=576746>Question 576746</A>: <I><SPAN STYLE=\"word-break: break-all;\"> GCF (a,b)=4 and LCM (a,b)=96<BR>\n" );
document.write( "Find all pairs of whole numbers a and b for which both statements are true, and show how to get them.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #369979 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=369979\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Since a and b are factors of 96, we can let\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a = 2^m3^n\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE b = 2^p3^q\">\r<BR>\n" );
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document.write( "Also note that 96 = (2^5)(3^1) and 4 = (2^2)(3^0). From the given information we can conclude that:\r<BR>\n" );
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document.write( "at least one of m or p is 5<BR>\n" );
document.write( "at least one of n or q is 1<BR>\n" );
document.write( "at least one of m or p is 2<BR>\n" );
document.write( "at least one of n or q is 0\r<BR>\n" );
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document.write( "The first two statements come from the fact that the LCM of two numbers a and b is computed by taking the highest exponent for each prime factor of a and b. Similar for GCF. However, we have two known values for m and p, and two known values for n and q. We can use casework to find all possible pairs {a,b}.\r<BR>\n" );
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document.write( "m = 5, n = 1 --> p = 2, q = 0 --> a = 96, b = 4<BR>\n" );
document.write( "m = 5, n = 0 --> p = 2, q = 1 --> a = 32, b = 12<BR>\n" );
document.write( "m = 2, n = 1 --> p = 5, q = 0 --> a = 12, b = 32<BR>\n" );
document.write( "m = 2, n = 0 --> p = 5, q = 1 --> a = 4, b = 96\r<BR>\n" );
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document.write( "Therefore the only ordered pairs {a,b} are {96,4}, {32,12}, {12,32}, and {4,96}. </SPAN>\n" );
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