document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=848383>Question 848383</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 4. Find the unit digit of 3^100. (The unit digit is the \ones digit\".) Hint: Use Euler's<BR>\n" );
document.write( "Theorem or Fermat's Little Theorem with p = 5, and remember that 3^100 will be an<BR>\n" );
document.write( "odd number.<BR>\n" );
document.write( "Questions are taken from the relevant sections of the textbook, An introduction<BR>\n" );
document.write( "to Abstract Algebra with notes to the future teacher (Nicodemi, et. al.). This<BR>\n" );
document.write( "week covers Sections 1.7, 2.1.<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 #511052 by </B><A HREF=/tutors/aboutme.mpl?userid=swincher4391><B>swincher4391(1107)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=swincher4391><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=swincher4391><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=511052\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The units digit of 3^100 is the remainder when dividing by 10.\r<BR>\n" );
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document.write( "So 3^100 (mod 10) is what we are looking for.\r<BR>\n" );
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document.write( "notice 3^2 = 9  = -1(mod 10)\r<BR>\n" );
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document.write( "3^(2n) = (-1)^n (mod 10)\r<BR>\n" );
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document.write( "3^(2*50) = (-1)^50 (mod 10)\r<BR>\n" );
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document.write( "Hence the remainder is 1.\r<BR>\n" );
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