document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=485869>Question 485869</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the greatest number of consecutive positive integers whose sum is 585? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #332217 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=332217\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We can say that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a + (a+1) + ... + (a+n-1) = 585\"> where a and n are integers. This simplifies to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE an + (1+2+...+(n-1)) = 585\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE an + \frac{(n-1)(n)}{2} = 585\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE n(a + \frac{n-1}{2}) = 585\">\r<BR>\n" );
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document.write( "If n = 585, then (n-1)/2 = 292 and we can set a = -291. We can check that (-291) + (-290) + ... + 291 + 292 + 293 = 585, because all the terms except 292 and 293 cancel (and 292 + 293 is obviously 585). Hence n = 585 is the greatest such number. </SPAN>\n" );
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