document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197922>Question 1197922</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many integer solutions (a,b,c) make the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2%2Bc%5E2=289 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2%2Bc%5E2=289\"   DISABLED_style= \"max-width:90%; height:auto;\" > true? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #831391 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=831391\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "If a,b,c are nonnegative integers where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3C=+b+%3C=+c ALIGN=MIDDLE  ALT=\"a+%3C=+b+%3C=+c\"   DISABLED_style= \"max-width:90%; height:auto;\" > then the only integer solutions to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2%2Bc%5E2+=+289 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2%2Bc%5E2+=+289\"   DISABLED_style= \"max-width:90%; height:auto;\" > are the following<BR>\n" );
document.write( "(a,b,c) = (0,0,17)<BR>\n" );
document.write( "(a,b,c) = (0,8,15)<BR>\n" );
document.write( "(a,b,c) = (1,12,12)<BR>\n" );
document.write( "(a,b,c) = (8,9,12)<BR>\n" );
document.write( "These four solutions are found through trial-and-error. Note that the second class (0,8,15) is from the pythagorean triple 8,15,17 which gives the equation 8^2+15^2 = 17^2 from the pythagorean theorem.\r<BR>\n" );
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document.write( "Let's remove the requirement that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3C=+b+%3C=+c ALIGN=MIDDLE  ALT=\"a+%3C=+b+%3C=+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "So we can permute the elements, meaning something like (0,0,17) leads to (0,17,0) and (17,0,0). There are 3 such permutations.\r<BR>\n" );
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document.write( "Let's now remove the requirement that the numbers a,b,c must be nonnegative.<BR>\n" );
document.write( "That yields twice as many solutions to give us 3*2 = 6 different solutions just based on the class (a,b,c) = (0,0,17). This is because the 17 could represent +17 or -17.\r<BR>\n" );
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document.write( "The class (0,8,15) has 3! = 3*2*1 = 6 permutations<BR>\n" );
document.write( "The 8 could be +8 or -8; the 15 could be +15 or -15<BR>\n" );
document.write( "So we multiply by 2*2 = 4 to get 4*6 = 24 different solutions from the class (a,b,c) = (0,8,15)\r<BR>\n" );
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document.write( "The class (1,12,12) has 3! = 6 permutations if we could tell the '12's apart; however we cannot. So we divide by 2 to fix this erroneous double-counting to get 6/2 = 3 permutations instead.<BR>\n" );
document.write( "Each item could be positive or negative, so we multiply by 2^3 = 2*2*2 = 8 to get 8*3 = 24 different solutions of the class (a,b,c) = (1,12,12)\r<BR>\n" );
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document.write( "Lastly, the class (a,b,c) = (8,9,12) has 3! = 6 permutations. Each shows up 2 times to get us 2*2*2*6 = 8*6 = 48 different solutions here.<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >class</TD><TD >count</TD></tr><tr><TD >0,0,17</TD><TD >6</TD></tr><tr><TD >0,8,15</TD><TD >24</TD></tr><tr><TD >1,12,12</TD><TD >24</TD></tr><tr><TD >8,9,12</TD><TD >48</TD></tr>\n" );
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document.write( "From here we add up the frequencies to get<BR>\n" );
document.write( "6+24+24+48 = 102<BR>\n" );
document.write( "which is the number of integer solutions to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2%2Bc%5E2+=+289 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2%2Bc%5E2+=+289\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Answer: 102<BR>\n" );
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