document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=27145>Question 27145</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the rightmost digit of 7^(1111). How do i find the remainder when divided by 10 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #15092 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=15092\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> WE ARE ONLY INTERESTED IN DIGIT IN UNITS PLACE.SO LET US TRY TO LOOK<BR>\n" );
document.write( "FOR A PATTERN IN THIS.<BR>\n" );
document.write( "POWER......UNITS DIGIT<BR>\n" );
document.write( "7^1=.......................7<BR>\n" );
document.write( "7^2=.......................9<BR>\n" );
document.write( "7^3=.......................3<BR>\n" );
document.write( "7^4-........................1<BR>\n" );
document.write( "-----------------------------<BR>\n" );
document.write( "7^5=......................7...SO IT REPEATS NOW..<BR>\n" );
document.write( "HENCE WE HAVE TO FIND 1111 POWER IS UNDER WHICH GROUP..SO DIVIDE IT<BR>\n" );
document.write( "WITH 4 AS THE ABOVE CYCLE REPEATS AFTER FREQUENCY OF 4.IT GIVES US A<BR>\n" );
document.write( "REMAINDER OF 3<BR>\n" );
document.write( "1111=4*277+3....SO IT FALLS UNDER GROUP OF 7^3...SO ITS UNITS DIGIT IS<BR>\n" );
document.write( "3 AS SHOWN ABOVE<BR>\n" );
document.write( "7^1111={7^(4*277)}{7^3)=IT HAS SAME ENDING AS 7^3 WHICH IS 3<BR>\n" );
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document.write( "IF YOU KNOW RESIDUAL CLASSES,CONGRUENCE MODULO,WE CAN USE THAT METHOD<BR>\n" );
document.write( "WE HAVE TO FIND X IN <BR>\n" );
document.write( "7^1111=X(MOD10)<BR>\n" );
document.write( "WE KNOW THAT<BR>\n" );
document.write( "7^2=49=-1(MOD10)<BR>\n" );
document.write( "RAISING TO 555 POWER BOTH SIDES,WE GET,<BR>\n" );
document.write( "(7^2)^555=(-1)^555(MOD10)<BR>\n" );
document.write( "7^1110=-1(MOD10)<BR>\n" );
document.write( "MULTIPLYING WITH 7 NOW WE GET <BR>\n" );
document.write( "7^1111=-7(MOD10)<BR>\n" );
document.write( "7^1111=(10-7)(MOD10)=3(MOD10)<BR>\n" );
document.write( "HENCE 3 IS THE REMAINDER WHEN DIVIDED WITH 10 OR UNITS PLACE HAS 3 <BR>\n" );
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