document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=339034>Question 339034</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If 713^10 is multiplied out completely, what is the units digit of the resulting number?\r<BR>\n" );
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document.write( "(a) 0 (b) 1 (c) 3 (d) 7 (e) 9 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #243027 by </B><A HREF=/tutors/aboutme.mpl?userid=galactus><B>galactus(183)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=galactus><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=galactus><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=243027\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> When looking for a units digit of a large number, use mod 10\r<BR>\n" );
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document.write( "Since a number divisible by 10 ends in 0, the units digit will be the remainder when divided by 10.\r<BR>\n" );
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document.write( "Note that 713=23*31\r<BR>\n" );
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document.write( "So, we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2823%5E2%29%5E5%2A%2831%5E2%29%5E5=713%5E10 ALIGN=MIDDLE  ALT=\"%2823%5E2%29%5E5%2A%2831%5E2%29%5E5=713%5E10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Powers of 31 always end in 1 and 23^2 ends in 9\r<BR>\n" );
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document.write( "23^2==9(mod 10) and 31==1(mod 10)\r<BR>\n" );
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document.write( "9*1=9\r<BR>\n" );
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document.write( "The last digit is 9.\r<BR>\n" );
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document.write( "The idea with mod arithmetic like this is to break it up into smaller powers to work with.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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