document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=111888>Question 111888</A>: <I><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "Could you please help me with this problem.  Thanks in advance.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "What is the remainder when 3 to the power 777777 is divided by 7.\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #81640 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=81640\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If you raise 3 to successive powers starting with 1, and then perform integer division on the result by 7, you get a repeating pattern of remainders that repeats every 6th time.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Use the modulo function which returns the remainder for an integer division: <BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3mod7=+++++++3 ALIGN=MIDDLE  ALT=\"3mod7=+++++++3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9mod7=%092 ALIGN=MIDDLE  ALT=\"9mod7=%092\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27mod7=%096 ALIGN=MIDDLE  ALT=\"27mod7=%096\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=81mod7=%094 ALIGN=MIDDLE  ALT=\"81mod7=%094\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=243mod7=%095 ALIGN=MIDDLE  ALT=\"243mod7=%095\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=729mod7=%091 ALIGN=MIDDLE  ALT=\"729mod7=%091\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2187mod7=%093 ALIGN=MIDDLE  ALT=\"2187mod7=%093\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6561mod7=%092 ALIGN=MIDDLE  ALT=\"6561mod7=%092\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19683mod7=%096 ALIGN=MIDDLE  ALT=\"19683mod7=%096\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=59049mod7=%094 ALIGN=MIDDLE  ALT=\"59049mod7=%094\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=177147mod7=%095 ALIGN=MIDDLE  ALT=\"177147mod7=%095\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16777216mod7=1 ALIGN=MIDDLE  ALT=\"16777216mod7=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now, 777777 mod 6 is 3 (mod 6 because it repeats every sixth time), so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E%28777777%29+mod+7 ALIGN=MIDDLE  ALT=\"3%5E%28777777%29+mod+7\"   DISABLED_style= \"max-width:90%; height:auto;\" > should yield the third item in the pattern, namely 6\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );