document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207613>Question 1207613</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For a positive integer $n$, $\phi(n)$ denotes the number of positive integers less than or equal to $n$ that are relatively prime to $n$.\r<BR>\n" );
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document.write( "What is $\phi(5)$? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845634 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=845634\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red>4</font>\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "For any prime p, we have the following:<BR>\n" );
document.write( "phi(p) = p-1\r<BR>\n" );
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document.write( "There are p-1 positive integers smaller than p that are relatively prime to p.<BR>\n" );
document.write( "1, 2, 3, ..., p-2, p-1<BR>\n" );
document.write( "This intuitively makes sense because all of these values are not factors of the prime p (well except for the trivial case 1)\r<BR>\n" );
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document.write( "Examples:<BR>\n" );
document.write( "phi(7) = 6<BR>\n" );
document.write( "phi(11) = 10\r<BR>\n" );
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document.write( "If we had some value not relatively prime to p, smaller than p, then it would mean p isn't prime.<BR>\n" );
document.write( "For instance if 2 and p weren't relatively prime, then p = 2k and p is even. But at this point p is not prime.\r<BR>\n" );
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document.write( "Further Reading<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://mathworld.wolfram.com/TotientFunction.html\">https://mathworld.wolfram.com/TotientFunction.html</a><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://en.wikipedia.org/wiki/Euler%27s_totient_function\">https://en.wikipedia.org/wiki/Euler%27s_totient_function</a><BR>\n" );
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