document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=944729>Question 944729</A>: <I><SPAN STYLE=\"word-break: break-all;\"> ONE NUMBER IN A PYTHAGOREAN TRIPLE IS 25, FIND THE OTHER TWO </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #576034 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=576034\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There is more than one answer to that.<BR>\n" );
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document.write( "A very popular Pythagorean triple (and the only one I remember) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%283%2C4%2C5%29%22 ALIGN=MIDDLE  ALT=\"%22%283%2C4%2C5%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "A (right) triangle with those measurements can be scaled up to another triangle whose measurements are other Pythagorean triples.<BR>\n" );
document.write( "with a scale factor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28%22%2815%2C20%2C25%29%22%29 ALIGN=MIDDLE  ALT=\"highlight%28%22%2815%2C20%2C25%29%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "A Pythagorean triple like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2815%2C20%2C25%29%22 ALIGN=MIDDLE  ALT=\"%22%2815%2C20%2C25%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > is considered a \"multiple\", while <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%283%2C4%2C5%29%22 ALIGN=MIDDLE  ALT=\"%22%283%2C4%2C5%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > is considered a \"primitive\".<BR>\n" );
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document.write( "A formula to generate the primitive Pythagorean triples is <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2mn ALIGN=MIDDLE  ALT=\"2mn\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2C%22 ALIGN=MIDDLE  ALT=\"%22%2C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2-n%5E2 ALIGN=MIDDLE  ALT=\"m%5E2-n%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2C%22 ALIGN=MIDDLE  ALT=\"%22%2C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2%2Bn%5E2 ALIGN=MIDDLE  ALT=\"m%5E2%2Bn%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > with positive integers <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%3En ALIGN=MIDDLE  ALT=\"m%3En\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "We can use that formula to generate another primitive Pythagorean triple that includes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2%2Bn%5E2=5 ALIGN=MIDDLE  ALT=\"m%5E2%2Bn%5E2=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28m=2%2Cn=1%29 ALIGN=MIDDLE  ALT=\"system%28m=2%2Cn=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > generates <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282mn=2%2A2%2A1=4%2Cm%5E2-n%5E2=2%5E2=1%5E2=4-1=3%2Cm%5E2%2Bn%5E2=2%5E2%2B1%5E2=4%2B1=5%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "which gives us the very popular <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%283%2C4%2C5%29%22 ALIGN=MIDDLE  ALT=\"%22%283%2C4%2C5%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We cannot make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2mn=5 ALIGN=MIDDLE  ALT=\"2mn=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but we can also make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2-n%5E2=5 ALIGN=MIDDLE  ALT=\"m%5E2-n%5E2=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28m%2Bn%29%2A%28m-n%29=5 ALIGN=MIDDLE  ALT=\"%28m%2Bn%29%2A%28m-n%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28m%2Bn=5%2Cm-n=1%29 ALIGN=MIDDLE  ALT=\"system%28m%2Bn=5%2Cm-n=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28m=3%2Cn=2%29 ALIGN=MIDDLE  ALT=\"system%28m=3%2Cn=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28m=3%2Cn=2%29 ALIGN=MIDDLE  ALT=\"system%28m=3%2Cn=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives us <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282mn=2%2A3%2A2=12%2Cm%5E2-n%5E2=3%5E2-2%5E2=9-4=5%2Cm%5E2%2Bn%5E2=3%5E2%2B2%5E2=9%2B4=13%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Then, primitive Pythagorean triple <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%285%2C12%2C13%29%22 ALIGN=MIDDLE  ALT=\"%22%285%2C12%2C13%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be scaled up by a factor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28%22%2825%2C60%2C65%29%22%29 ALIGN=MIDDLE  ALT=\"highlight%28%22%2825%2C60%2C65%29%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "We can also use the formula to generate another primitive Pythagorean triple that includes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25 ALIGN=MIDDLE  ALT=\"25\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We cannot make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2mn=25 ALIGN=MIDDLE  ALT=\"2mn=25\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2-n=%28m%2Bn%29%28m-n%29=25 ALIGN=MIDDLE  ALT=\"m%5E2-n=%28m%2Bn%29%28m-n%29=25\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We can only make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2%2Bn%5E2=25 ALIGN=MIDDLE  ALT=\"m%5E2%2Bn%5E2=25\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that requires <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28m=4%2Cn=3%29 ALIGN=MIDDLE  ALT=\"system%28m=4%2Cn=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2%2Bn%5E2=25 ALIGN=MIDDLE  ALT=\"m%5E2%2Bn%5E2=25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28m=4%2Cn=3%29 ALIGN=MIDDLE  ALT=\"system%28m=4%2Cn=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > generates <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282mn=2%2A4%2A3=24%2Cm%5E2-n%5E2=4%5E2-3%5E2=16-9=7%2Cm%5E2%2Bn%5E2=4%5E2%2B3%5E2=16%2B9=25%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That gives us <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28%22%287%2C24%2C25%29%22%29 ALIGN=MIDDLE  ALT=\"highlight%28%22%287%2C24%2C25%29%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
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