document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1046143>Question 1046143</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I didn't ask the type the question right before<BR>\n" );
document.write( "Find the smallest odd positive integer (m) so that the decimal expansion of 1/m has just 4 non recurring digits and has 2 repeating digits.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #661702 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5345)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=661702\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F99=%220.010101...%22 ALIGN=MIDDLE  ALT=\"1%2F99=%220.010101...%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F99=%220.020202...%22 ALIGN=MIDDLE  ALT=\"2%2F99=%220.020202...%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and for any <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > integer that is not a multiple of 11, or 99,<BR>\n" );
document.write( "the decimal part of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2F99 ALIGN=MIDDLE  ALT=\"p%2F99\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be a 2-digit repeating sequence.\r<BR>\n" );
document.write( "\n" );
document.write( "So, we can represent all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%5E4%2Fm ALIGN=MIDDLE  ALT=\"10%5E4%2Fm\"   DISABLED_style= \"max-width:90%; height:auto;\" > numbers whose decimal expansion is a 2-digit repeating sequence as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2F99 ALIGN=MIDDLE  ALT=\"p%2F99\"   DISABLED_style= \"max-width:90%; height:auto;\" > , where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" >= an integer that is not a multiple of 11.<BR>\n" );
document.write( "Then, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%5E4%2Fm=p%2F99 ALIGN=MIDDLE  ALT=\"10%5E4%2Fm=p%2F99\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fm=%281%2F10%5E4%29%28p%2F99%29 ALIGN=MIDDLE  ALT=\"1%2Fm=%281%2F10%5E4%29%28p%2F99%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "has pairs of repeating digits after the 4th decimal place.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fm=%281%2F10%5E4%29%28p%2F99%29 ALIGN=MIDDLE  ALT=\"1%2Fm=%281%2F10%5E4%29%28p%2F99%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=99%2A10%5E4%2Fp ALIGN=MIDDLE  ALT=\"m=99%2A10%5E4%2Fp\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=3%5E2%2A11%2A2%5E4%2A5%5E4%2Fp ALIGN=MIDDLE  ALT=\"m=3%5E2%2A11%2A2%5E4%2A5%5E4%2Fp\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be an integer,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > cannot have any factors not present in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E2%2A11%2A2%5E4%2A5%5E4 ALIGN=MIDDLE  ALT=\"3%5E2%2A11%2A2%5E4%2A5%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be an odd integer,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > must have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E4 ALIGN=MIDDLE  ALT=\"2%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a factor.<BR>\n" );
document.write( "Choosing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=3%5E2%2A2%5E4%2A5%5E4 ALIGN=MIDDLE  ALT=\"p=3%5E2%2A2%5E4%2A5%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=3%5E2%2A11%2A2%5E4%2A5%5E4%2F%283%5E2%2A2%5E4%2A5%5E4%29=11 ALIGN=MIDDLE  ALT=\"m=3%5E2%2A11%2A2%5E4%2A5%5E4%2F%283%5E2%2A2%5E4%2A5%5E4%29=11\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fm=%220.09090909...%22 ALIGN=MIDDLE  ALT=\"1%2Fm=%220.09090909...%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is the smallest odd positive integer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28m%29 ALIGN=MIDDLE  ALT=\"%28m%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > so that the decimal expansion of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fm ALIGN=MIDDLE  ALT=\"1%2Fm\"   DISABLED_style= \"max-width:90%; height:auto;\" > has pairs of repeating digits after the 4th decimal place.<BR>\n" );
document.write( "However, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=11 ALIGN=MIDDLE  ALT=\"m=11\"   DISABLED_style= \"max-width:90%; height:auto;\" > may not be considered the right answer,<BR>\n" );
document.write( "because the phrase \"just 4 non recurring digits\" probably means that the third decimal places must be different from the 5th one,<BR>\n" );
document.write( "and the and fourth one must be different from the sixth one.<BR>\n" );
document.write( "So, we need to put less factors into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "leaving <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > with more of the factors in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E2%2A11%2A5%5E4 ALIGN=MIDDLE  ALT=\"3%5E2%2A11%2A5%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=3%2A11=33 ALIGN=MIDDLE  ALT=\"m=3%2A11=33\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fm=1%2F33=%220.03030303...%22 ALIGN=MIDDLE  ALT=\"1%2Fm=1%2F33=%220.03030303...%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > (not right either),<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=5%2A11=55 ALIGN=MIDDLE  ALT=\"m=5%2A11=55\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fm=1%2F33=%220.02020202m=...%22 ALIGN=MIDDLE  ALT=\"1%2Fm=1%2F33=%220.02020202m=...%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > (not right either),<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=3%5E2%2A11=99 ALIGN=MIDDLE  ALT=\"m=3%5E2%2A11=99\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fm=1%2F99=%220.01010101...%22 ALIGN=MIDDLE  ALT=\"1%2Fm=1%2F99=%220.01010101...%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and so on.<BR>\n" );
document.write( "We could keep trying to drop factors from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "until we get to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=11%2A5%5E6=6875 ALIGN=MIDDLE  ALT=\"m=11%2A5%5E6=6875\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fm=1%2F6875=%220.0001454545...%22 ALIGN=MIDDLE  ALT=\"1%2Fm=1%2F6875=%220.0001454545...%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "has a 4th decimal digit <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29 ALIGN=MIDDLE  ALT=\"%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > that is different from the sixth one <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%29 ALIGN=MIDDLE  ALT=\"%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28m=6875%29 ALIGN=MIDDLE  ALT=\"highlight%28m=6875%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is my answer.<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE: there is an option that does not require trying so many factor combinations to get to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but it does require more algebra, so it may not get to the answer faster. </SPAN>\n" );
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