document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=307268>Question 307268</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the units digit of 3^35?\r<BR>\n" );
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document.write( "a  1  b  3   c  5   d  7   e  9<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 #219850 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=219850\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The units digit repeats every 4 times.\r<BR>\n" );
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document.write( "3^0 = 1<BR>\n" );
document.write( "3^1 = 3<BR>\n" );
document.write( "3^2 = 9<BR>\n" );
document.write( "3^3 = 27\r<BR>\n" );
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document.write( "3^4 = 81<BR>\n" );
document.write( "3^5 = 243<BR>\n" );
document.write( "3^6 = 729<BR>\n" );
document.write( "3^7 = 2187\r<BR>\n" );
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document.write( "3^8 = 6561<BR>\n" );
document.write( "3^9 = 19683<BR>\n" );
document.write( "3^10 = 59049<BR>\n" );
document.write( "3^11 = 177147\r<BR>\n" );
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document.write( "3^12 = 531441<BR>\n" );
document.write( "3^13 = 1594323<BR>\n" );
document.write( "3^14 = 4782969<BR>\n" );
document.write( "3^15 = 1438907\r<BR>\n" );
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document.write( "3^16 = 43046721<BR>\n" );
document.write( "3^17 = 129140163<BR>\n" );
document.write( "3^18 = 387420489<BR>\n" );
document.write( "3^19 = 1162261467\r<BR>\n" );
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document.write( "If you divide the exponent by 4, then the remainder will tell you what the equivalent base exponent is.\r<BR>\n" );
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document.write( "The base exponents are 0,1,2,3.\r<BR>\n" );
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document.write( "Using the base exponents, you get:\r<BR>\n" );
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document.write( "3^0 = 1 which gives you a units digit of 1.<BR>\n" );
document.write( "3^1 = 3 which gives you a units digit of 3.<BR>\n" );
document.write( "3^2 = 9 which gives you a units digit of 9.<BR>\n" );
document.write( "3^3 = 27 which gives you a units digit of 7.\r<BR>\n" );
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document.write( "We'll run some tests with numbers we can verify and then extrapolate from that to numbers we can't verify easily.\r<BR>\n" );
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document.write( "We'll start with exponent of 16 and work up to 19.\r<BR>\n" );
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document.write( "16/4 = 4 with a remainder of 0.<BR>\n" );
document.write( "3^0 = 1<BR>\n" );
document.write( "3^16 = 43046721.<BR>\n" );
document.write( "Units digit is confirmed to be equal to 1.\r<BR>\n" );
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document.write( "17/4 = 4 with a remainder of 1.<BR>\n" );
document.write( "3^1 = 3<BR>\n" );
document.write( "3^17 = 129140163<BR>\n" );
document.write( "Units digit is confirmed to be equal to 3.\r<BR>\n" );
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document.write( "18/4 = 4 with a remainder of 2.<BR>\n" );
document.write( "3^2 = 9<BR>\n" );
document.write( "3^18 = 387420489<BR>\n" );
document.write( "Units digit is confirmed to be equal to 9.\r<BR>\n" );
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document.write( "19/4 = 4 with a remainder of 3.<BR>\n" );
document.write( "3^3 = 27<BR>\n" );
document.write( "3^19 = 1162261467<BR>\n" );
document.write( "Units digit is confirmed to be equal to 7.\r<BR>\n" );
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document.write( "Looks like our formula is good so we use it to find the units digit of a number we can't verify so easily.\r<BR>\n" );
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document.write( "We will now find the units digit for 3^35 using our formula.\r<BR>\n" );
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document.write( "35/4 = 8 with a remainder of 3.\r<BR>\n" );
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document.write( "Our base exponent is 3.\r<BR>\n" );
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document.write( "3^3 = 27\r<BR>\n" );
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document.write( "The units digit for 3^35 equals 7.\r<BR>\n" );
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document.write( "That would be selection d.\r<BR>\n" );
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