document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=856598>Question 856598</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the smallest natural number that can be expressed as the sum of the squares of two, not necessarily distinct, natural numbers in two different ways.<BR>\n" );
document.write( "Show all work!\r<BR>\n" );
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document.write( "Thank you!<BR>\n" );
document.write( "-Lulu </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #516201 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=516201\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> My answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2850%29 ALIGN=MIDDLE  ALT=\"highlight%2850%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , but my work is not that elegant.<BR>\n" );
document.write( "I would post the problem in the forum of the artofproblemsolving website,<BR>\n" );
document.write( "and then try to understand the very brief and elegant answer obtained there.\r<BR>\n" );
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document.write( "A pair of natural numbers <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > (not necessarily different from each other, maybe <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=b ALIGN=MIDDLE  ALT=\"a=b\"   DISABLED_style= \"max-width:90%; height:auto;\" > ) squared ad up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "A different pair of natural numbers <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > (not necessarily different from each other) squared ad up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2%2Bd%5E2=a%5E2%2Bb%5E2 ALIGN=MIDDLE  ALT=\"c%5E2%2Bd%5E2=a%5E2%2Bb%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "One way would be making a table with<BR>\n" );
document.write( "values for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > (from 1 to 10) in the first column,<BR>\n" );
document.write( "values for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > (from 1 to 10) in the first row,<BR>\n" );
document.write( "and the sum of squares <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where column <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > meets row <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Comparing each values on each row, to other values<BR>\n" );
document.write( "on the rows below, we would find many matching sums of squares of the same pair of numbers.<BR>\n" );
document.write( "They are symmetrically arranged to either side of the diagonal.<BR>\n" );
document.write( "We would be looking for matches between sums of squares of different pairs of numbers,<BR>\n" );
document.write( "like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%5E2%2B1%5E2=49%2B1=50 ALIGN=MIDDLE  ALT=\"7%5E2%2B1%5E2=49%2B1=50\"   DISABLED_style= \"max-width:90%; height:auto;\" > matching <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5E2%2B5%5E2=25%2B25=50 ALIGN=MIDDLE  ALT=\"5%5E2%2B5%5E2=25%2B25=50\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%5E2%2B1%5E2=64%2B1=65 ALIGN=MIDDLE  ALT=\"8%5E2%2B1%5E2=64%2B1=65\"   DISABLED_style= \"max-width:90%; height:auto;\" > matching <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%5E2%2B4%5E2=49%2B16=65 ALIGN=MIDDLE  ALT=\"7%5E2%2B4%5E2=49%2B16=65\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Another approach:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2%2Bd%5E2=a%5E2%2Bb%5E2 ALIGN=MIDDLE  ALT=\"c%5E2%2Bd%5E2=a%5E2%2Bb%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2-a%5E2=b%5E2-d%5E2 ALIGN=MIDDLE  ALT=\"c%5E2-a%5E2=b%5E2-d%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28c%2Ba%29%28c-a%29=%28b%2Bd%29%28b-d%29 ALIGN=MIDDLE  ALT=\"%28c%2Ba%29%28c-a%29=%28b%2Bd%29%28b-d%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So we need an integer that can be written as two different products,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28c%2Ba%29%28c-a%29 ALIGN=MIDDLE  ALT=\"%28c%2Ba%29%28c-a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2Bd%29%28b-d%29 ALIGN=MIDDLE  ALT=\"%28b%2Bd%29%28b-d%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%2A1=5%2A3 ALIGN=MIDDLE  ALT=\"15%2A1=5%2A3\"   DISABLED_style= \"max-width:90%; height:auto;\" > would lead you to solve systems like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28c%2Ba=15%2Cc-a=1%29 ALIGN=MIDDLE  ALT=\"system%28c%2Ba=15%2Cc-a=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28c=8%2Ca=7%29 ALIGN=MIDDLE  ALT=\"system%28c=8%2Ca=7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28b%2Bd=5%2Cb-d=3%29 ALIGN=MIDDLE  ALT=\"system%28b%2Bd=5%2Cb-d=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28b=4%2Cd=1%29 ALIGN=MIDDLE  ALT=\"system%28b=4%2Cd=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28c%5E2%2Bd%5E2=8%5E2%2B1%5E2=64%2B1-65%2Ca%5E2%2Bb%5E2=7%5E2%2B4%5E2=49%2B16=65%29 ALIGN=MIDDLE  ALT=\"system%28c%5E2%2Bd%5E2=8%5E2%2B1%5E2=64%2B1-65%2Ca%5E2%2Bb%5E2=7%5E2%2B4%5E2=49%2B16=65%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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