document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=164864>Question 164864</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the ones's digit of 8^1007?\r<BR>\n" );
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document.write( "THANK YOU SO MUCH! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #121494 by </B><A HREF=/tutors/aboutme.mpl?userid=gonzo><B>gonzo(654)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=gonzo><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=121494\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> an interesting thing happens to the number as the exponent gets past 16.<BR>\n" );
document.write( "the ones digit becomes 0 and stays 0 thereafter.<BR>\n" );
document.write( "as you go up in exponent power, the number of 0's on the right hand side of the number keeps expanding.<BR>\n" );
document.write( "for example: <BR>\n" );
document.write( "8^17 has one zero on the right (unit digit).<BR>\n" );
document.write( "8^18 has 3 zeroes on the right (unit, ten, hundred digit).<BR>\n" );
document.write( "8^22 has 5 zeroes on the right (unit, ten, hundred, thousand, ten thousand digit)<BR>\n" );
document.write( "so, even though my calculator and my excel spreadsheet don't go up that far, i would assume the ones digit will remain 0 after 8^17. <BR>\n" );
document.write( "for a while (up to 8^11), the ones digit was repeating (8,4,2,6,8,4,2,6,8,4,2,6) and i was tempted to give you that answer but my curiosity got the best of me after my calculator threw up on 8^12 and i went to the excel spreadsheet.  it was there that i saw what was really happening.<BR>\n" );
document.write( "here's a display of what the excel spreadsheet was telling me.<BR>\n" );
document.write( "the first column is the number raised to the power in the second column.<BR>\n" );
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document.write( "x	:	8 raised to the power of x<BR>\n" );
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document.write( "1	:	8<BR>\n" );
document.write( "2	:	64<BR>\n" );
document.write( "3	:	512<BR>\n" );
document.write( "4	:	4096<BR>\n" );
document.write( "5	:	32768<BR>\n" );
document.write( "6	:	262144<BR>\n" );
document.write( "7	:	2097152<BR>\n" );
document.write( "8	:	16777216<BR>\n" );
document.write( "9	:	134217728<BR>\n" );
document.write( "10	:	1073741824<BR>\n" );
document.write( "11	:	8589934592<BR>\n" );
document.write( "12	:	68719476736<BR>\n" );
document.write( "13	:	549755813888<BR>\n" );
document.write( "14	:	4398046511104<BR>\n" );
document.write( "15	:	35184372088832<BR>\n" );
document.write( "16	:	281474976710656<BR>\n" );
document.write( "17	:	2251799813685250<BR>\n" );
document.write( "18	:	18014398509482000<BR>\n" );
document.write( "19	:	144115188075856000<BR>\n" );
document.write( "20	:	1152921504606850000<BR>\n" );
document.write( "21	:	9223372036854780000<BR>\n" );
document.write( "22	:	73786976294838200000<BR>\n" );
document.write( "23	:	590295810358706000000<BR>\n" );
document.write( "24	:	4722366482869650000000<BR>\n" );
document.write( "25	:	37778931862957200000000<BR>\n" );
document.write( "26	:	302231454903657000000000<BR>\n" );
document.write( "27	:	2417851639229260000000000<BR>\n" );
document.write( "28	:	19342813113834100000000000<BR>\n" );
document.write( "29	:	154742504910673000000000000<BR>\n" );
document.write( "30	:	1237940039285380000000000000<BR>\n" );
document.write( "31	:	9903520314283040000000000000<BR>\n" );
document.write( "32	:	79228162514264300000000000000<BR>\n" );
document.write( "33	:	633825300114115000000000000000<BR>\n" );
document.write( "34	:	5070602400912920000000000000000<BR>\n" );
document.write( "35	:	40564819207303300000000000000000<BR>\n" );
document.write( "36	:	324518553658427000000000000000000<BR>\n" );
document.write( "37	:	2596148429267410000000000000000000<BR>\n" );
document.write( "38	:	20769187434139300000000000000000000<BR>\n" );
document.write( "39	:	166153499473114000000000000000000000<BR>\n" );
document.write( "40	:	1329227995784920000000000000000000000<BR>\n" );
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document.write( "i'm not sure what the algebraic answer to this would be.<BR>\n" );
document.write( "i would supposes that after a certain amount of raising to a power for any number that the ones digit will eventually start being 0, but i don't know how to explain that other than it happens.  for example: 2 raised to the 50th power starts exhibiting the same thing, i.e. the ones digit becomes 0 and remains 0 thereafter.<BR>\n" );
document.write( "a similarity in both these instances is that the number of digits other than 0 maxed at around 15.<BR>\n" );
document.write( "did the same thing with 3 to the power of whatever and max number of digits before started getting 0's on the right appeared to be 15 as well.<BR>\n" );
document.write( "i'm sure there's an explanation.<BR>\n" );
document.write( "i just don't know it.<BR>\n" );
document.write( "anyway, 0 is your answer to the best of my knowledge.<BR>\n" );
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