document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=255255>Question 255255</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the one’s digit of the number 7^2005 ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #187527 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=187527\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 7^1 = 7<BR>\n" );
document.write( "7^2 = 49<BR>\n" );
document.write( "7^3 = 343<BR>\n" );
document.write( "7^4 = 2401<BR>\n" );
document.write( "7^5 = 16807<BR>\n" );
document.write( "7^6 = 117649<BR>\n" );
document.write( "7^7 = 823543<BR>\n" );
document.write( "7^8 = 5764801<BR>\n" );
document.write( "7^9 = 40353607<BR>\n" );
document.write( "7^10 = 282475249<BR>\n" );
document.write( "7^11 = 1977326743\r<BR>\n" );
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document.write( "the last digit is repeating in a pattern of:\r<BR>\n" );
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document.write( "7,9,3,1,<BR>\n" );
document.write( "7,9,3,1,<BR>\n" );
document.write( "7,9,3,1,\r<BR>\n" );
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document.write( "to find out what the last digit is, you need to make a formula.\r<BR>\n" );
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document.write( "the formula would be taking the larger exponent and breaking it down into the smaller exponent\r<BR>\n" );
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document.write( "example:\r<BR>\n" );
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document.write( "suppose your number is 7^6.\r<BR>\n" );
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document.write( "6/4 = 2 and 7^2 = 49 so the last digit is 9.\r<BR>\n" );
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document.write( "look at 7^6 above and you can see that the last digit is 9.\r<BR>\n" );
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document.write( "suppose your number is 7^11.\r<BR>\n" );
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document.write( "11/4 = 2 with a remainder of 3.\r<BR>\n" );
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document.write( "7^3 = 343 so the last digit is 3.\r<BR>\n" );
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document.write( "look up 7^11 above and you can see that the last digit is 3.\r<BR>\n" );
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document.write( "you can use this formula for larger numbers even though you can't check them because the calculator doesn't carry enough digits.\r<BR>\n" );
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document.write( "I used excel and the highest I could go was 7^17 as shown below:\r<BR>\n" );
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document.write( "base	exponent	number	           last digit\r\n" );
document.write( "7	1	        7	           7\r\n" );
document.write( "7	2	       49	           9\r\n" );
document.write( "7	3	       343	           3\r\n" );
document.write( "7	4	       2401	           1\r\n" );
document.write( "7	5	       16807	           7\r\n" );
document.write( "7	6	       117649	           9\r\n" );
document.write( "7	7	       823543	           3\r\n" );
document.write( "7	8	       5764801	           1\r\n" );
document.write( "7	9	       40353607	           7\r\n" );
document.write( "7	10	       282475249	   9\r\n" );
document.write( "7	11	       1977326743	   3\r\n" );
document.write( "7	12	       13841287201	   1\r\n" );
document.write( "7	13	       96889010407	   7\r\n" );
document.write( "7	14	       678223072849	   9\r\n" );
document.write( "7	15	       4747561509943	   3\r\n" );
document.write( "7	16	       33232930569601	   1\r\n" );
document.write( "7	17	       232630513987207	   7\r\n" );
document.write( "7       18                                 9\r\n" );
document.write( "7       19                                 3\r\n" );
document.write( "7       20                                 1 \r\n" );
document.write( "7       21			           etc.....        \r\n" );
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document.write( "for a final check, try 7^16.\r<BR>\n" );
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document.write( "16/4 = 4 with a remainder of 0.\r<BR>\n" );
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document.write( "7^0 = 1 so the remainder should be 1.\r<BR>\n" );
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document.write( "it is, as shown in the table from excel above.\r<BR>\n" );
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document.write( "note that 16 / 4 = 4 with a remainder of 0.   it also equals 3 with a remainder of 4 which confirms that the last digit is 1.\r<BR>\n" );
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document.write( "In your problem, the number is 7^2005\r<BR>\n" );
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document.write( "divide 2005 by 4 to get 501 with a remainder of 1.\r<BR>\n" );
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document.write( "7^1 = 7, so the last digit of 7^2005 = 1\r<BR>\n" );
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document.write( " </SPAN>\n" );
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