document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1138427>Question 1138427</A>: <I><SPAN STYLE=\"word-break: break-all;\"> DETERMINE ALL THE TRIPLES OF POSITIVE INTEGERS (a,b,c), SUCH THAT ab+bc+ca= abc, WHERE A IS LESS THAN B LESS THAN C </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #756218 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=756218\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Given: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab%2Bac%2Bbc+=+abc ALIGN=MIDDLE  ALT=\"ab%2Bac%2Bbc+=+abc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Move one of the terms on the left to the right side.  It doesn't matter which one, because the original equation is symmetric in a, b, and c.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abc-ab+=+ac%2Bbc ALIGN=MIDDLE  ALT=\"abc-ab+=+ac%2Bbc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab%28c-1%29+=+c%28a%2Bb%29 ALIGN=MIDDLE  ALT=\"ab%28c-1%29+=+c%28a%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab%2F%28a%2Bb%29+=+c%2F%28c-1%29 ALIGN=MIDDLE  ALT=\"ab%2F%28a%2Bb%29+=+c%2F%28c-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That equation says that ab is an integer that is 1 more than (a+b).  So<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab+=+a%2Bb%2B1 ALIGN=MIDDLE  ALT=\"ab+=+a%2Bb%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now solve this equation for either variable in terms of the other.  Again it doesn't matter which, because again this equation is symmetric in a and b.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab-a+=+b%2B1 ALIGN=MIDDLE  ALT=\"ab-a+=+b%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28b-1%29+=+b%2B1 ALIGN=MIDDLE  ALT=\"a%28b-1%29+=+b%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+%28b%2B1%29%2F%28b-1%29 ALIGN=MIDDLE  ALT=\"a+=+%28b%2B1%29%2F%28b-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+%28%28b-1%29%2B2%29%2F%28b-1%29 ALIGN=MIDDLE  ALT=\"a+=+%28%28b-1%29%2B2%29%2F%28b-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+1%2B2%2F%28b-1%29 ALIGN=MIDDLE  ALT=\"a+=+1%2B2%2F%28b-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "1 is an integer, and a has to be an integer.  That means 2/(b-1) has to be an integer; and that means (b-1) has to be a factor of 2.<br><BR>\n" );
document.write( "So b-1 has to be either 1 or 2; that means b has to be either 2 or 3.<br><BR>\n" );
document.write( "And now we can find all the triples a, b, and c for which the given equation is true.<br><BR>\n" );
document.write( "(1) If b = 2 then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+1%2B2%2F1+=+1%2B2+=+3 ALIGN=MIDDLE  ALT=\"a+=+1%2B2%2F1+=+1%2B2+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "and then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ab%29%2F%28a%2Bb%29+=+c%2F%28c-1%29+=+6%2F5 ALIGN=MIDDLE  ALT=\"%28ab%29%2F%28a%2Bb%29+=+c%2F%28c-1%29+=+6%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means c is 6.<br><BR>\n" );
document.write( "The three numbers a, b, and c (in no particular order) are 2, 3, and 6.<br><BR>\n" );
document.write( "(2) If b = 3 then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+1%2B2%2F2+=+1%2B1+=+2 ALIGN=MIDDLE  ALT=\"a+=+1%2B2%2F2+=+1%2B1+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "and then (as before)<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ab%29%2F%28a%2Bb%29+=+c%2F%28c-1%29+=+6%2F5 ALIGN=MIDDLE  ALT=\"%28ab%29%2F%28a%2Bb%29+=+c%2F%28c-1%29+=+6%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means c is 6.<br><BR>\n" );
document.write( "So there is a single set of three integers for which the given equation is true.<br><BR>\n" );
document.write( "Finally, since the problem specifies a < b < c, the single solution is<br><BR>\n" );
document.write( "{a,b,c) = (2,3,6) </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );