document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1162750>Question 1162750</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If the positive integers a and b satisfy √a − √b = √11, what is the maximum value of a/b? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #786585 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13215)</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=786585\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The following original response left out important parts of the solution -- though the final answer to the question was correct.  See further down for the corrected/expanded response.<br><BR>\n" );
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document.write( "If a and b are integers and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%29-sqrt%28b%29+=+sqrt%2811%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%29-sqrt%28b%29+=+sqrt%2811%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then for some positive integer value of n,<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=%28n%2B1%29%5E2 ALIGN=MIDDLE  ALT=\"a=%28n%2B1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+n%5E2 ALIGN=MIDDLE  ALT=\"b+=+n%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fb+=+%28n%2B1%29%5E2%2Fn%5E2+=+%28%28n%2B1%29%2Fn%29%5E2 ALIGN=MIDDLE  ALT=\"a%2Fb+=+%28n%2B1%29%5E2%2Fn%5E2+=+%28%28n%2B1%29%2Fn%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "For positive integer values of n, the maximum value of a/b is clearly when n=1 and a/b = (2/1)^2 = 4.<br><BR>\n" );
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document.write( "Corrected response....<br><BR>\n" );
document.write( "If a and b are integers and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%29-sqrt%28b%29+=+sqrt%2811%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%29-sqrt%28b%29+=+sqrt%2811%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then for some positive integer value of n,<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=%2811%28n%2B1%29%5E2%29 ALIGN=MIDDLE  ALT=\"a=%2811%28n%2B1%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+11%28n%5E2%29 ALIGN=MIDDLE  ALT=\"b+=+11%28n%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "With those values of a and b,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%29+=+%28n%2B1%29%2Asqrt%2811%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%29+=+%28n%2B1%29%2Asqrt%2811%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28b%29+=+%28n%29%2Asqrt%2811%29 ALIGN=MIDDLE  ALT=\"sqrt%28b%29+=+%28n%29%2Asqrt%2811%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%29-sqrt%28b%29+=+%28n%2B1%29%2Asqrt%2811%29-%28n%29%2Asqrt%2811%29+=+sqrt%2811%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%29-sqrt%28b%29+=+%28n%2B1%29%2Asqrt%2811%29-%28n%29%2Asqrt%2811%29+=+sqrt%2811%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fb+=+%28n%2B1%29%5E2%2Fn%5E2+=+%28%28n%2B1%29%2Fn%29%5E2+=+%281%2B1%2Fn%29%5E2 ALIGN=MIDDLE  ALT=\"a%2Fb+=+%28n%2B1%29%5E2%2Fn%5E2+=+%28%28n%2B1%29%2Fn%29%5E2+=+%281%2B1%2Fn%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "For positive integer values of n, the maximum value of a/b is clearly when n=1 and a/b = (2/1)^2 = 4.<br><BR>\n" );
document.write( "CHECK....<br><BR>\n" );
document.write( "n=1: a = 11(2^2) = 44; b = 11(1^2) = 11; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%29-sqrt%28b%29+=+2%2Asqrt%2811%29-1%2Asqrt%2811%29+=+sqrt%2811%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%29-sqrt%28b%29+=+2%2Asqrt%2811%29-1%2Asqrt%2811%29+=+sqrt%2811%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >; a/b = 44/11 = 4<br><BR>\n" );
document.write( "n=2: a = 11(3^2) = 99; b = 11(2^2) = 44; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%29-sqrt%28b%29+=+3%2Asqrt%2811%29-2%2Asqrt%2811%29+=+sqrt%2811%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%29-sqrt%28b%29+=+3%2Asqrt%2811%29-2%2Asqrt%2811%29+=+sqrt%2811%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >; a/b = 99/44 = 9/4<br><BR>\n" );
document.write( "n=3: a = 11(4^2) = 176; b = 11(3^2) = 99; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%29-sqrt%28b%29+=+4%2Asqrt%2811%29-3%2Asqrt%2811%29+=+sqrt%2811%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%29-sqrt%28b%29+=+4%2Asqrt%2811%29-3%2Asqrt%2811%29+=+sqrt%2811%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >; a/b = 176/99 = 16/9<br><br><BR>\n" );
document.write( "Clearly for larger values of n the ratio a/b will continue to get smaller....<br><BR>\n" );
document.write( " </SPAN>\n" );
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