document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=464047>Question 464047</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. what is the smallest integers that can be express as the sum of the square of the two positive integers in two ways?<BR>\n" );
document.write( "2. fins two different numbers A and B such that a+b = (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 #318732 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=318732\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1. Suppose a,b,c,d are positive integers satisfying\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a^2 + b^2 = c^2 + d^2 = k\">\r<BR>\n" );
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document.write( "Then,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a^2 - c^2 = d^2 - b^2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (a-c)(a+c) = (d-b)(d+b)\">\r<BR>\n" );
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document.write( "After this was some \"educated trial-and-error.\" I simply thought of numbers that would factor in two different ways (e.g. 48 = 12*4 = 24*2). The only constraints were that the factors (e.g. 12,4 and 24,2) had to be both odd or both even, and distinct, so that a,b,c,d would be nonnegative integers. I obtained 15 = 15*1 = 5*3, which gives the equations\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a+c = 15\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a-c = 1\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE d-b = 5\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE d+b = 1\">\r<BR>\n" );
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document.write( "Solving this yields a = 8, c = 7, b = 1, d = 3, and it can be checked that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 8^2 + 1^2 = 7^2 + 4^2 = 65\">\r<BR>\n" );
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document.write( "which is the smallest such number (I hope).\r<BR>\n" );
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document.write( "2. Given\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a+b = ab\"> we can find b in terms of a,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a = ab - b = b(a-1)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE b = \frac{a}{a-1}\">\r<BR>\n" );
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document.write( "This essentially defines a function b in terms of a, and there are infinitely many ordered pairs (a,b) that satisfy. For example, if a = 3, b = 3/2, and 3 + (3/2) = 3(3/2) = 9/2. </SPAN>\n" );
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