document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1138622>Question 1138622</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all positive integers a and b so that (a+1)/b and (b+2)/a are simultaneously<BR>\n" );
document.write( "positive integers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #756565 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</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=756565\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Plan of attack:<br><BR>\n" );
document.write( "Given the requirement <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2B1%29%2Fb ALIGN=MIDDLE  ALT=\"%28a%2B1%29%2Fb\"   DISABLED_style= \"max-width:90%; height:auto;\" > equal to an integer, choose integer values for a and determine the possible integer values for b; for those possible values of b, determine whether <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2B2%29%2Fa ALIGN=MIDDLE  ALT=\"%28b%2B2%29%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also an integer.<br><BR>\n" );
document.write( "If a = 1...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2B1%29%2Fb+=+2%2Fb ALIGN=MIDDLE  ALT=\"%28a%2B1%29%2Fb+=+2%2Fb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The possible values for b are 1 and 2.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2B2%29%2Fa+=+%28b%2B2%29%2F1 ALIGN=MIDDLE  ALT=\"%28b%2B2%29%2Fa+=+%28b%2B2%29%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an integer for both these values of b.<BR>\n" );
document.write( "For b=1, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2B2%29%2Fa+=+3%2F1+=+3 ALIGN=MIDDLE  ALT=\"%28b%2B2%29%2Fa+=+3%2F1+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >  -->  (a,b) = (1,1) is a solution<BR>\n" );
document.write( "For b=2, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2B2%29%2Fa+=+4%2F1+=+4 ALIGN=MIDDLE  ALT=\"%28b%2B2%29%2Fa+=+4%2F1+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >  -->  (a,b) = (1,2) is a solution<br><BR>\n" );
document.write( "If a = 2...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2B1%29%2Fb+=+3%2Fb ALIGN=MIDDLE  ALT=\"%28a%2B1%29%2Fb+=+3%2Fb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The possible values for b are 1 and 3.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2B2%29%2Fa+=+%28b%2B2%29%2F2 ALIGN=MIDDLE  ALT=\"%28b%2B2%29%2Fa+=+%28b%2B2%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not an integer for either b=1 or b=3.<br><BR>\n" );
document.write( "If a = 3...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2B1%29%2Fb+=+4%2Fb ALIGN=MIDDLE  ALT=\"%28a%2B1%29%2Fb+=+4%2Fb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The possible values for b are 1, 2, and 4.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2B2%29%2Fa+=+%28b%2B2%29%2F3 ALIGN=MIDDLE  ALT=\"%28b%2B2%29%2Fa+=+%28b%2B2%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an integer for both b=1 and b=4.<BR>\n" );
document.write( "For b=1, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2B2%29%2Fa+=+3%2F3+=+1 ALIGN=MIDDLE  ALT=\"%28b%2B2%29%2Fa+=+3%2F3+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >  -->  (a,b) = (3,1) is a solution<BR>\n" );
document.write( "For b=4, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2B2%29%2Fa+=+6%2F3+=+2 ALIGN=MIDDLE  ALT=\"%28b%2B2%29%2Fa+=+6%2F3+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  -->  (a,b) = (3,4) is a solution<br><BR>\n" );
document.write( "If a = 4...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2B1%29%2Fb+=+5%2Fb ALIGN=MIDDLE  ALT=\"%28a%2B1%29%2Fb+=+5%2Fb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The possible values for b are 1 and 5.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2B2%29%2Fa+=+%28b%2B2%29%2F4 ALIGN=MIDDLE  ALT=\"%28b%2B2%29%2Fa+=+%28b%2B2%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not an integer for either b=1 or b=5.<br><BR>\n" );
document.write( "If a = 5...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2B1%29%2Fb+=+6%2Fb ALIGN=MIDDLE  ALT=\"%28a%2B1%29%2Fb+=+6%2Fb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The possible values for b are 1, 2, 3, and 6.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2B2%29%2Fa+=+%28b%2B2%29%2F5 ALIGN=MIDDLE  ALT=\"%28b%2B2%29%2Fa+=+%28b%2B2%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an integer for b=3.<BR>\n" );
document.write( "For b=3, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2B2%29%2Fa+=+5%2F5+=+1 ALIGN=MIDDLE  ALT=\"%28b%2B2%29%2Fa+=+5%2F5+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >  -->  (a,b) = (5,3) is a solution<br><BR>\n" );
document.write( "-----------------------------------------------------------------<br><BR>\n" );
document.write( "I haven't been able to come up with a formal proof; but it is clear to me that there will be no solutions with values of a greater than 5.  So the complete set of ordered pairs of positive integers (a,b) for which both <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2B1%29%2Fb ALIGN=MIDDLE  ALT=\"%28a%2B1%29%2Fb\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2B2%29%2Fa ALIGN=MIDDLE  ALT=\"%28b%2B2%29%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > are both integers is<BR>\n" );
document.write( "(1,1); (1,2); (3,1); (3,4); (5,3)<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );