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 #851508 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851508\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step by step.\r<BR>\n" );
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document.write( "**1. Prime Factorization**\r<BR>\n" );
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document.write( "* $6^6 = (2 \cdot 3)^6 = 2^6 \cdot 3^6$<BR>\n" );
document.write( "* $10^{10} = (2 \cdot 5)^{10} = 2^{10} \cdot 5^{10}$<BR>\n" );
document.write( "* $15^{15} = (3 \cdot 5)^{15} = 3^{15} \cdot 5^{15}$<BR>\n" );
document.write( "* $30^{30} = (2 \cdot 3 \cdot 5)^{30} = 2^{30} \cdot 3^{30} \cdot 5^{30}$\r<BR>\n" );
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document.write( "**2. Understanding Divisors**\r<BR>\n" );
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document.write( "A divisor of any of these numbers will be of the form $2^a \cdot 3^b \cdot 5^c$, where:\r<BR>\n" );
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document.write( "* $0 \le a \le 30$ (since $2^{30}$ is the highest power of 2)<BR>\n" );
document.write( "* $0 \le b \le 30$ (since $3^{30}$ is the highest power of 3)<BR>\n" );
document.write( "* $0 \le c \le 30$ (since $5^{30}$ is the highest power of 5)\r<BR>\n" );
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document.write( "**3. Total Possible Divisors**\r<BR>\n" );
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document.write( "If we consider all possible combinations of a, b, and c, we have:\r<BR>\n" );
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document.write( "* 31 choices for a (0 to 30)<BR>\n" );
document.write( "* 31 choices for b (0 to 30)<BR>\n" );
document.write( "* 31 choices for c (0 to 30)\r<BR>\n" );
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document.write( "Therefore, the total number of possible divisors is $31 \cdot 31 \cdot 31 = 31^3 = 29791$.\r<BR>\n" );
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document.write( "**4. Inclusion-Exclusion Principle**\r<BR>\n" );
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document.write( "We need to find the number of divisors that are divisors of *at least one* of the given numbers. We can use the inclusion-exclusion principle.\r<BR>\n" );
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document.write( "* Let A be the set of divisors of $6^6$.<BR>\n" );
document.write( "* Let B be the set of divisors of $10^{10}$.<BR>\n" );
document.write( "* Let C be the set of divisors of $15^{15}$.<BR>\n" );
document.write( "* Let D be the set of divisors of $30^{30}$.\r<BR>\n" );
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document.write( "We want to find $|A \cup B \cup C \cup D|$.\r<BR>\n" );
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document.write( "Since $30^{30}$ contains all prime factors of the other 3, any divisor of $6^6$, $10^{10}$, or $15^{15}$ will also be a divisor of $30^{30}$.<BR>\n" );
document.write( "Therefore, $|A \cup B \cup C \cup D| = |D|$.\r<BR>\n" );
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document.write( "* $|D| = (30+1)(30+1)(30+1) = 31^3 = 29791$\r<BR>\n" );
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document.write( "**5. Final Answer**\r<BR>\n" );
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document.write( "The number of positive integers that are divisors of at least one of the given numbers is $31^3 = 29791$.\r<BR>\n" );
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document.write( "**Therefore, the answer is 29791.**<BR>\n" );
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