document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=548989>Question 548989</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the least positive integral value of n for which (n - 12) divided by <BR>\n" );
document.write( "(5n + 23) is a non-zero reducible fraction?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #358234 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=358234\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Assume n > 12 so that the fraction is positive (1 through 11 don't work anyway). If the fraction is reducible for some n, then n-12 and 5n+23 must have a common factor p (other than 1). We can write this using modular arithmetic:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE n-12 \equiv 5n + 23 \equiv 0\"> (mod p)\r<BR>\n" );
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document.write( "This implies <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE n \equiv 12\"> (mod p), so <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 5n + 23 \equiv 5(12) + 23 \equiv 83 \equiv 0\"> (mod p). Hence, p = 83. Therefore, we can set n = 95, obtaining the fraction 83/498, or 1/6.\r<BR>\n" );
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