document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=18069>Question 18069</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am working with prime fractions such as 1/7, 2/7, 3/7......6/7. and my teacher asked me to make a \"clock diagram\" I don't know what the hec he is talking about...please help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #10383 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><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=10383\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  1/7, 2/7, 3/7......6/7. and my teacher asked me to make a \"clock diagram\"\r<BR>\n" );
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document.write( " 1/7 = 0.142857<BR>\n" );
document.write( " 2/7 = 0.285714<BR>\n" );
document.write( " 3/7 = 0.428571<BR>\n" );
document.write( " 4/7 = 0.571428<BR>\n" );
document.write( " 5/7 = 0.714285<BR>\n" );
document.write( " 6/7 = 0.857142\r<BR>\n" );
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document.write( " 7 is a very special number, with 1/7 is a rational with (unterminatng) <BR>\n" );
document.write( " decimal of period 6. In fact, 1/7 = 0.[0.142857] (cyclic of period 6)<BR>\n" );
document.write( " Also, other fractions i/(2 <=i <= 6) are also cyclic in these 6 numbers<BR>\n" );
document.write( " without changing the order, as you can see above)\r<BR>\n" );
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document.write( " In group theory, they froma cyclic group of order 6, namely<BR>\n" );
document.write( " Z7*,*) ,where Z7* = {[0],[1],[2],[3],[4],[5],[6]}\r<BR>\n" );
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document.write( " I am not quite sure about \"Clock diagram\"<BR>\n" );
document.write( " One way you can try is to draw a clockwise rotation (regular hexagon with<BR>\n" );
document.write( " vetices marked in the order as)  <BR>\n" );
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document.write( " 142857 --> 428571 --> 285714 --> 857142 -->571428 --> 714285<BR>\n" );
document.write( "  1/7  --> 3/7    -->  2/7  --> 6/7 --> 4/7  --> 5/7 \r<BR>\n" );
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document.write( " Like   1/7<BR>\n" );
document.write( "  *3  /   \*3<BR>\n" );
document.write( " 5/7 /      \ 3/7 <BR>\n" );
document.write( "     |       |<BR>\n" );
document.write( "   *3|       |*3<BR>\n" );
document.write( "    4/7      2/7<BR>\n" );
document.write( "      \    /<BR>\n" );
document.write( "    *3 \  /*3<BR>\n" );
document.write( "        6/7  <BR>\n" );
document.write( " [Sorry for hard to have good diagram here.]     <BR>\n" );
document.write( " Note 3/7 = (10*1/7 -1), 2/7 =(10 * 3/7 -4)  , 6/7 =(10* 2/7 -2) ,<BR>\n" );
document.write( " 4/7 =(10* 6/7 -8) , 5/7 =(10* 4/7 -5)<BR>\n" );
document.write( " If you know modulus(remainder dividing by 7)<BR>\n" );
document.write( ", you can see 1/7 * 3 = 3/7, 3/7*3 = 9/7 = 2/7,<BR>\n" );
document.write( " 2/7 * 3 = 6/7, 6/7 * 3 = 18/7 = 4/7, 4/7 * 3= 12/7 = 5/7 and 5/7*3 = 15/7 = 1/7.<BR>\n" );
document.write( " (this means each fraction is 3 times ofthe previous one.<BR>\n" );
document.write( " I.E. (i+1) mod 7/ 7 = (3* i) mod 7/7 for all 1<=i<= 5.\r<BR>\n" );
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document.write( " Good luck !!\r<BR>\n" );
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document.write( " Kenny<BR>\n" );
document.write( "   </SPAN>\n" );
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