document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207622>Question 1207622</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let $m$ be a positive integer.  If $m$ has exactly $18$ positive divisors, then how many positive divisors does $m^2$ have? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845608 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=845608\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The prime p has exactly two divisors: 1 and itself.<BR>\n" );
document.write( "Example: the prime 7 has factors 1 and 7 only.\r<BR>\n" );
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document.write( "p^k has k+1 divisors and they are:<BR>\n" );
document.write( "p^0, p^1, p^2, ..., p^(k-1), p^k<BR>\n" );
document.write( "Note that p^0 = 1 and p^1 = p.<BR>\n" );
document.write( "Example: p = 3 and k = 4 yields p^k = 3^4 = 81 having k+1 = 4+1 = 5 divisors. <BR>\n" );
document.write( "Those five divisors are 1, 3, 9, 27, 81.\r<BR>\n" );
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document.write( "Let p and q represent two different primes.<BR>\n" );
document.write( "Let k and m be positive integers.<BR>\n" );
document.write( "We can construct the following composite number<BR>\n" );
document.write( "n = p^k*q^m\r<BR>\n" );
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document.write( "How many divisors does n have?<BR>\n" );
document.write( "Imagine we can form a table that has k+1 rows and m+1 columns.<BR>\n" );
document.write( "Along the left hand side we will have the labels p^0, p^1, p^2, ..., p^(k-1), p^k.<BR>\n" );
document.write( "Along the top header we will have the labels q^0, q^1, q^2, ..., q^(m-1), q^m.\r<BR>\n" );
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document.write( "This table generates (k+1)*(m+1) different divisors.<BR>\n" );
document.write( "Each entry into the table is the result of multiplying the headers. Eg: multiply p^2 with q^3 to get p^2q^3 as one divisor of n.\r<BR>\n" );
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document.write( "This logic can be extended to as many unique prime factors as you want.<BR>\n" );
document.write( "Example: n = p^k*q^m*r^s will have (k+1)*(m+1)*(s+1) divisors.<BR>\n" );
document.write( "We add 1 to each exponent, then multiply those results.\r<BR>\n" );
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document.write( "More info found here<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://mathworld.wolfram.com/DivisorFunction.html\">https://mathworld.wolfram.com/DivisorFunction.html</a><BR>\n" );
document.write( "Refer specifically to equation (3) on that page.\r<BR>\n" );
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document.write( "So this is why tutor greenestamps had written various ways to factor 18. <BR>\n" );
document.write( "His goal is to write an equation like<BR>\n" );
document.write( "18 = (k+1)(m+1)<BR>\n" );
document.write( "or<BR>\n" );
document.write( "18 = (k+1)(m+1)(s+1)\r<BR>\n" );
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document.write( "Let me know if you have any questions.<BR>\n" );
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