document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=476158>Question 476158</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Does there exist a permutation a1, a2, . . . , a8 of the numbers 1, 2, . . . , 8, such that the 8 numbers a1, a1+a2, . . . , a1+a2+...+a8 all leave a di&#1092;&#1092;erent remainder when divided<BR>\n" );
document.write( "by 8? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #326722 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=326722\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Yes. Set a_1 = 8, a_k = k-1 for k > 1. To simplify things, we can actually set a_1 = 0 since this will not change the remainder mod 8. The sums will be 0, 1, 3, 6, 10, 15, 21, 28 (all triangular numbers except for 0). These numbers mod 8 are 0, 1, 3, 6, 2, 7, 5, 4, all different.\r<BR>\n" );
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document.write( "Note that you can actually prove that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a_1 \equiv 0 (mod 8)\"> by taking the differences between successive terms like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (a_1 + a_2) - (a_1) = a_2 \not\equiv 0 (mod 8)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (a_1 + a_2 + a_3) - (a_1 + a_2) = a_3 \not\equiv 0 (mod 8)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (a_1 + ... + a_8) - (a_1 + ... + a_7) = a_8 \not\equiv 0 (mod 8)\">\r<BR>\n" );
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document.write( "Since a_2, ..., a_8 cannot be 0 mod 8, it follows that a_1 = 8. </SPAN>\n" );
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