document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1150159>Question 1150159</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many different perfect squares are factors of 2007^2007? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #771517 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=771517\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
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document.write( "(corrected response, thanks to tutor @ikleyn for pointing out the obvious error....)<BR>\n" );
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document.write( "The sum of the digits of 2007 is 9, so the number is divisible by 9:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2007+=+3%2A3%2A223 ALIGN=MIDDLE  ALT=\"2007+=+3%2A3%2A223\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "223 is prime, so we have the prime factorization of 2007:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2007+=+%283%5E2%29%28223%5E1%29 ALIGN=MIDDLE  ALT=\"2007+=+%283%5E2%29%28223%5E1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2007%5E2007+=+%283%5E4014%29%28223%5E2007%29 ALIGN=MIDDLE  ALT=\"2007%5E2007+=+%283%5E4014%29%28223%5E2007%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A perfect square factor of the number has to be of the form<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%5Ea%29%28223%5Eb%29 ALIGN=MIDDLE  ALT=\"%283%5Ea%29%28223%5Eb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Where a and b are even integers, with 0 <= a <= 4014 and 0 <= b <= 2007.<br><BR>\n" );
document.write( "That gives us 2008 choices for a and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%282007%29 ALIGN=MIDDLE  ALT=\"cross%282007%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > 1004 choices for b; the number of perfect square factors of the number is then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%282008%2A2007+=+4030056%29 ALIGN=MIDDLE  ALT=\"cross%282008%2A2007+=+4030056%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > 2008*1004 = 2016032.<br>\r<BR>\n" );
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