document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210186>Question 1210186</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the number of positive integers that are divisors of at least one of 6^{6}, 10^{10}, 15^{15}, and 30^{30}.<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 #851545 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</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=851545\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Note that 6^6, 10^10, and 15^15 are all factors of 30^30.  It is therefore only necessary to find the number of positive integers that are factors of 30^30.<br><BR>\n" );
document.write( "The standard method for determining the number of factors of a given integer N is<BR>\n" );
document.write( "(1) find the prime factorization of N;<BR>\n" );
document.write( "(2) add 1 to each exponent in the prime factorization; and<BR>\n" );
document.write( "(3) multiply the numbers from step (2)<br><BR>\n" );
document.write( "The prime factorization of 30 is 2*3*5, so the prime factorization of 30^30 is (2^30)(3^30)(5^30).<br><BR>\n" );
document.write( "The number of factors of 30^30 is (30+1)(30+1)(30+1) = 31^3 = 29791<br><BR>\n" );
document.write( "ANSWER: 29791<br><BR>\n" );
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