document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=408264>Question 408264</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show 5n+3 and 7n+4 are relatively prime for all n </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #287832 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=287832\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> As a corollary to Euclid's algorithm, there are integers s and t such that as + bt = gcd(a,b).  Hence if we're able to find integers s,t, such that \r<BR>\n" );
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document.write( "s(5n + 3) + t(7n + 4) = 1, \r<BR>\n" );
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document.write( "then we've shown that 5n + 3, 7n + 4 are relatively prime for all n.\r<BR>\n" );
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document.write( "==> (5s + 7t)n + (3s + 4t) = 1.\r<BR>\n" );
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document.write( "It's enough to see if the system <BR>\n" );
document.write( "5s + 7t = 0<BR>\n" );
document.write( "3s + 4t = 1<BR>\n" );
document.write( "has integer solutions.<BR>\n" );
document.write( "Indeed, the solutions are s = 7, and t = -5.<BR>\n" );
document.write( "Therefore 5n + 3, 7n + 4 are relatively prime for all n. </SPAN>\n" );
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