document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1108845>Question 1108845</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let n be a three digit number and let m be the number obtained by reversing the order of the digits in n. Suppose that m does not equal n and that n+m and n-m are both divisible by 7. Find all such pairs n and m.\r<BR>\n" );
document.write( "\n" );
document.write( "I have found <BR>\n" );
document.write( "168 and 861<BR>\n" );
document.write( "259 and 952\r<BR>\n" );
document.write( "\n" );
document.write( "Are there any more?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #723872 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</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=723872\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are no other solutions. My proof is below.<BR>\n" );
document.write( "The kids at artofproblemsolving would read it and laugh.<BR>\n" );
document.write( "They would probably say that it was obvious based on the Ruszcyk theorem,<BR>\n" );
document.write( "or some such thing.<BR>\n" );
document.write( "(I just made that theorem up, but they probably have a few modular arithmetic theorems).<BR>\n" );
document.write( " <BR>\n" );
document.write( " <BR>\n" );
document.write( "Let the numbers be made of the digits <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=100a%2B10b%2Bc ALIGN=MIDDLE  ALT=\"n=100a%2B10b%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=100c%2B10b%2Ba ALIGN=MIDDLE  ALT=\"m=100c%2B10b%2Ba\"   DISABLED_style= \"max-width:90%; height:auto;\" > are 3-digit numbers, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C%3E0 ALIGN=MIDDLE  ALT=\"a%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%3C%3E0 ALIGN=MIDDLE  ALT=\"c%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "To have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3C%3Em ALIGN=MIDDLE  ALT=\"n%3C%3Em\"   DISABLED_style= \"max-width:90%; height:auto;\" > , it must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C%3Ec ALIGN=MIDDLE  ALT=\"a%3C%3Ec\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and to have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-m%3E0 ALIGN=MIDDLE  ALT=\"n-m%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , it must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3Ec ALIGN=MIDDLE  ALT=\"a%3Ec\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "For any two numbers x and y,<BR>\n" );
document.write( "if x-y and x+7 are both multiples of 7,<BR>\n" );
document.write( "x and y are both multiples of 7,<BR>\n" );
document.write( "and vice versa.<BR>\n" );
document.write( "So, m and n must be multiples of 7.<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=100a%2B10b%2Bc=%287%2A14%2B2%29a%2B%287%2B3%29b%2Bc=7%2814a%2Bb%29%2B%282a%2B3b%2Bc%29 ALIGN=MIDDLE  ALT=\"n=100a%2B10b%2Bc=%287%2A14%2B2%29a%2B%287%2B3%29b%2Bc=7%2814a%2Bb%29%2B%282a%2B3b%2Bc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be a multiple of 7,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%2B3b%2Bc ALIGN=MIDDLE  ALT=\"2a%2B3b%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be a multiple of 7.<BR>\n" );
document.write( "Similarly,  for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=100c%2B10b%2Ba ALIGN=MIDDLE  ALT=\"m=100c%2B10b%2Ba\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be a multiple of 7,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2c%2B3b%2Ba ALIGN=MIDDLE  ALT=\"2c%2B3b%2Ba\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be a multiple of 7.<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%2B3b%2Bc ALIGN=MIDDLE  ALT=\"2a%2B3b%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2c%2B3b%2Ba ALIGN=MIDDLE  ALT=\"2c%2B3b%2Ba\"   DISABLED_style= \"max-width:90%; height:auto;\" > are both multiples of 7,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282a%2B3b%2Bc%29-%282c%2B3b%2Ba%29=a-c ALIGN=MIDDLE  ALT=\"%282a%2B3b%2Bc%29-%282c%2B3b%2Ba%29=a-c\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be a multiple of 7, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282a%2B3b%2Bc%29%2B%282c%2B3b%2Ba%29=3%28a%2Bc%29%2B6b=3%28a%2Bc%2B2b%29 ALIGN=MIDDLE  ALT=\"%282a%2B3b%2Bc%29%2B%282c%2B3b%2Ba%29=3%28a%2Bc%29%2B6b=3%28a%2Bc%2B2b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be a multiple of 7,<BR>\n" );
document.write( "which means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bc%2B2b ALIGN=MIDDLE  ALT=\"a%2Bc%2B2b\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be a multiple of 7.<BR>\n" );
document.write( " <BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-c ALIGN=MIDDLE  ALT=\"a-c\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be a multiple of 7, the only choices are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28c=1%2Ca=1%2B7=8%29 ALIGN=MIDDLE  ALT=\"system%28c=1%2Ca=1%2B7=8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28c=2%2Ca=2%2B7=9%29 ALIGN=MIDDLE  ALT=\"system%28c=2%2Ca=2%2B7=9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28c=1%2Ca=8%29 ALIGN=MIDDLE  ALT=\"system%28c=1%2Ca=8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bc=9 ALIGN=MIDDLE  ALT=\"a%2Bc=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and then for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bc%2B2b=9%2B2b ALIGN=MIDDLE  ALT=\"a%2Bc%2B2b=9%2B2b\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be a multiple of 7,<BR>\n" );
document.write( "it must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2B2b=21 ALIGN=MIDDLE  ALT=\"9%2B2b=21\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=6 ALIGN=MIDDLE  ALT=\"b=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So one solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a=8%2Cb=6%2Cc=1%29 ALIGN=MIDDLE  ALT=\"system%28a=8%2Cb=6%2Cc=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28n=861%2Cm=168%29 ALIGN=MIDDLE  ALT=\"system%28n=861%2Cm=168%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28c=2%2Ca=9%29 ALIGN=MIDDLE  ALT=\"system%28c=2%2Ca=9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bc=11 ALIGN=MIDDLE  ALT=\"a%2Bc=11\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and then for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bc%2B2b=11%2B2b ALIGN=MIDDLE  ALT=\"a%2Bc%2B2b=11%2B2b\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be a multiple of 7,<BR>\n" );
document.write( "it must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2B2b=21 ALIGN=MIDDLE  ALT=\"11%2B2b=21\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=5 ALIGN=MIDDLE  ALT=\"b=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So the other solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a=9%2Cb=5%2Cc=2%29 ALIGN=MIDDLE  ALT=\"system%28a=9%2Cb=5%2Cc=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28n=952%2Cm=259%29 ALIGN=MIDDLE  ALT=\"system%28n=952%2Cm=259%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );