document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207719>Question 1207719</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let $m$ and $n$ be non-negative integers.  If $m = 6n + 2$, then what integer between $0$ and $m$ is the inverse of $2$ modulo $m$?  Answer in terms of $n$. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845760 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=845760\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll follow a similar line of thinking as the other tutor, but show a slightly different pathway.\r<BR>\n" );
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document.write( "Let x be some integer in the set {0,1,2,3,...,m-2,m-1} such that 2x = 1 (mod m), assuming such a value exists.<BR>\n" );
document.write( "x is the multiplicative inverse of 2 in mod m\r<BR>\n" );
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document.write( "This converts to 2x-1 = km for some integer k.<BR>\n" );
document.write( "The general rule is that if a = b (mod n) then a-b is a multiple of n, i.e. a-b = kn for some integer k. <BR>\n" );
document.write( "Rephrased: a & b generate the same remainder when dividing by n.\r<BR>\n" );
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document.write( "That slight tangent aside, let's revisit 2x-1 = km to apply a substitution.\r<BR>\n" );
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document.write( "2x-1 = k*m<BR>\n" );
document.write( "2x-1 = k(6n+2) ......... plug in m = 6n + 2<BR>\n" );
document.write( "2x-1 = 2k(3n+1)<BR>\n" );
document.write( "2x - 2k(3n+1) = 1<BR>\n" );
document.write( "2(x - k(3n+1) ) = 1<BR>\n" );
document.write( "2(some integer) = 1\r<BR>\n" );
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document.write( "The left hand side on the last line is always even, but the right hand side is odd. This contradiction proves that there aren't any integer solutions to 2x-1 = km where m = 6n+2.<BR>\n" );
document.write( "By extension, there aren't any solutions to 2x = 1 (mod m) either.<BR>\n" );
document.write( "2 does not have a multiplicative inverse in mod m.\r<BR>\n" );
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document.write( "Answer: No solutions.<BR>\n" );
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