document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=182270>Question 182270</A>: <I><SPAN STYLE=\"word-break: break-all;\"> At hicksville high school, the students celebrate a very unusual tradition.  You see there are exactly 100 students at HHS and exactly 100 lockers.  Every november 23, the students celebrate what has come to be known as \"Locker Day\".  Each of the 100 students line up and number off from 1 to 100.  Student number 1 goes through the hallways and opens each of the 100 lockers. student number 2 then goes through and closes every other locker. Student number 3 follows and \"reverses\" every third locker.  The fourth then goes through and \"reverses\" every fourth locker.  This continues untill each of the 100 students has passed through the hallways of HHS, ending with the 100th student reversing the 100th locker.  My question for you: After this is done, which of the lockers will remain open?<BR>\n" );
document.write( "I need an equation and the answer to the question above.<BR>\n" );
document.write( "so far i have<BR>\n" );
document.write( "1---4---9---16---25---36---49---64---81---100<BR>\n" );
document.write( "-+3--+5--+7---+9---+11--+13--+15--+17---+19\r<BR>\n" );
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document.write( "thanks for taking the time for reading this question and maybe even helping me answer it<BR>\n" );
document.write( "P.S. this is only a math problem not a real life situation at my school<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 #136813 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=136813\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "All integers except for perfect squares have an even number of even divisors, including 1 and the integer itself.  Perfect squares have an odd number of divisors because the integer square root is a divisor.\r<BR>\n" );
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document.write( "Examples:\r<BR>\n" );
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document.write( "6 has 4 factors: 1, 2, 3, and 6.  Not a perfect square, even number of factors.\r<BR>\n" );
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document.write( "16 has 5 factors: 1, 2, 4, 8, 16. Perfect square, odd number of factors.\r<BR>\n" );
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document.write( "You could get all fancy and prove this fact by saying something along the lines of:  If <i>p</i> is an even divisor of integer <i>n</i>, then there is an integer quotient <i>q</i> that is also a divisor, hence divisors come in pairs.  Except for the case of a perfect square where <i>p</i> is the square root of <i>n</i> in which case the quotient is also <i>p</i>, making the exception to the 'divisors in pairs' rule. \r<BR>\n" );
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document.write( "If Student number <i>s</i> touches locker <i>n</i>, then <i>s</i> must be an even divisor of <i>n</i>, according to the rules of the game.\r<BR>\n" );
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document.write( "Therefore, non-perfect square number lockers are touched an even number of times.  That is, if they started out closed, they end up closed.  Perfect square number lockers are touched an odd number of times -- if they started closed, they end up open.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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