document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=42055>Question 42055</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Express 1994 as a sum of coonsecutive positive integers and show this is the only way to do it. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #27190 by </B><A HREF=/tutors/aboutme.mpl?userid=kev82><B>kev82(151)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=kev82><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=27190\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hi,\r<BR>\n" );
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document.write( "1994 can be written as 497+498+499+500 so there is how it's done. Proving this is the only way is quite simple but it's quite long and I don't really want to type it here. An outline of the proof is as follows.\r<BR>\n" );
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document.write( "1) To write a number N as the sum of (possibly negative) consecutive integers, where the first term is a. Then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N=na%2Bn%28n-1%29%2F2 ALIGN=MIDDLE  ALT=\"N=na%2Bn%28n-1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "2) We can rearrage to find a\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=%282N-n%28n-1%29%29%2F2n ALIGN=MIDDLE  ALT=\"a=%282N-n%28n-1%29%29%2F2n\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "3) We want a to be an integer so (replace equals with congruent to)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n-1%29=2N+mod+2n ALIGN=MIDDLE  ALT=\"n%28n-1%29=2N+mod+2n\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "4) (Be careful with 2N congruent 0 mod 2n here!) If n is odd then this has a solution when n divides N. If n is even then there is solution when n does not divide N, but does divide 2N.\r<BR>\n" );
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document.write( "5) So taking N=1994, the only values of n that satisfy the above conditions are n=4,997,3988. (N=1 is obviously a trivial solution, but you really want n>1 right?)\r<BR>\n" );
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document.write( "6) Calculating a for the above 3 values gives 497, -496, and -1993. You are only interested in the sums which use strictly posotive numbers. There is only one of those, with n=4 which I gave you above.\r<BR>\n" );
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document.write( "Hope that helps,<BR>\n" );
document.write( "Kev </SPAN>\n" );
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