document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=913080>Question 913080</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help with my homework. How do I solve this:<BR>\n" );
document.write( "What is the units digit of 13^162 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #554283 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=554283\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> List out the powers of 13\r<BR>\n" );
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document.write( "13^0 = 1<BR>\n" );
document.write( "13^1 = 13<BR>\n" );
document.write( "13^2 = 169<BR>\n" );
document.write( "13^3 = 2,197<BR>\n" );
document.write( "13^4 = 28,561<BR>\n" );
document.write( "13^5 = 371,293\r<BR>\n" );
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document.write( "Notice the units digits (from top to bottom) are: 1, 3, 9, 7, 1, 3, ...\r<BR>\n" );
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document.write( "So we repeat the units digit once we get to 13^4. Effectively, this tells us that we repeat every 4 increases of the exponent in 13^x (x is an integer).\r<BR>\n" );
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document.write( "We can take advantage of this by dividing the exponent by 4 and looking at the remainder. For instance, 5/4 = 1 remainder 1. That remainder 1 matches up with the 1 in the exponent of 13^1\r<BR>\n" );
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document.write( "In other words, 13^5 and 13^1 have the same units digit (due to that remainder 1)\r<BR>\n" );
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document.write( "Now do 162/4 = 40 remainder 2\r<BR>\n" );
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document.write( "This tells us 13^(162) and 13^2 have the same units digit.\r<BR>\n" );
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document.write( "Therefore, the final answer is <font size=4 color=\"red\">9</font> (since 13^2 = 169 has a units digit of 9).\r<BR>\n" );
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document.write( "Let me know if you need more help or if you need me to explain a step in more detail.<BR>\n" );
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document.write( "Thanks,\r<BR>\n" );
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document.write( "Jim  </SPAN>\n" );
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