document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194009>Question 1194009</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Choose all the descriptions for natural numbers n that have 3 divisors.\r<BR>\n" );
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document.write( "1- n= p^2 * q (for any two distinct primes p and q)<BR>\n" );
document.write( "2- n= p * q     (for any two distinct primes p and q)<BR>\n" );
document.write( "3- n= p * q * r (for any three distinct primes p,q and r)<BR>\n" );
document.write( "4- n= p^2 (for any prime number)\r<BR>\n" );
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document.write( "So every natural number has at least 2 factors - 1 and itself. So numbers with 3 factors then have to be perfect squares of prime numbers.\r<BR>\n" );
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document.write( "I selected 4- n= p^2 (for any prime number)\r<BR>\n" );
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document.write( "So they only have 1 distinct prime factor, and the question says select ALL.<BR>\n" );
document.write( "Am I missing any other description that applies?\r<BR>\n" );
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document.write( "Thanks\r<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 #826090 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</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=826090\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Let's focus on n = p^2*q for now.\r<BR>\n" );
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document.write( "The unique primes are p and q<BR>\n" );
document.write( "Though we have p show up twice in the form p^2\r<BR>\n" );
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document.write( "The divisors are:<BR>\n" );
document.write( "1, p, q, p*q, p^2, p^2q<BR>\n" );
document.write( "Clearly there are more than 3 divisors here so we move on.\r<BR>\n" );
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document.write( "With n = p*q, the divisors are: <BR>\n" );
document.write( "1, p, q, p*q<BR>\n" );
document.write( "We get close to 3 divisors, but we have one too many.\r<BR>\n" );
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document.write( "Now onto n = p*q*r<BR>\n" );
document.write( "Divisors: 1, p, q, r, p*q, p*r, q*r, p*q*r<BR>\n" );
document.write( "There are 8 divisors here, which we rule this answer choice out as well.<BR>\n" );
document.write( "Thing to notice: There are 3 atomic pieces of p,q,r so there are 2^3 = 8 different divisors. <BR>\n" );
document.write( "Why does this work? I'll leave it for you to think about. Hint: Think of the power set.\r<BR>\n" );
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document.write( "Lastly n = p^2<BR>\n" );
document.write( "The divisors are: 1, p, p^2<BR>\n" );
document.write( "This is exactly 3 divisors, so you have chosen the correct answer. This is the only answer that fits the 3 divisors pattern.\r<BR>\n" );
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document.write( "In summary, you are correct to think choice 4 is the only answer. The quick reasoning is that squaring any prime will have exactly 3 factors.\r<BR>\n" );
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document.write( "Some concrete numeric examples:\r<BR>\n" );
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document.write( "For n = p^2*q we could pick p = 5 and q = 7<BR>\n" );
document.write( "n = 5^2*7 = 175<BR>\n" );
document.write( "Divisors: 1, 5, 7, 25, 35, 175<BR>\n" );
document.write( "Number of divisors: 6\r<BR>\n" );
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document.write( "Now onto the form n = p*q<BR>\n" );
document.write( "Let's go with p = 11 and q = 13<BR>\n" );
document.write( "n = 11*13 = 143<BR>\n" );
document.write( "Factors: 1, 11, 13, 143<BR>\n" );
document.write( "Number of divisors: 4\r<BR>\n" );
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document.write( "Form: n = p*q*r<BR>\n" );
document.write( "Let p = 2, q = 3, r = 7<BR>\n" );
document.write( "n = 2*3*7 = 42<BR>\n" );
document.write( "Divisors: 1, 2, 3, 6, 7, 14, 21, 42<BR>\n" );
document.write( "Number of divisors: 8\r<BR>\n" );
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document.write( "Form: n = p^2<BR>\n" );
document.write( "Let p = 17<BR>\n" );
document.write( "n = p^2 = 17^2 = 289<BR>\n" );
document.write( "Divisors: 1, 17, 289<BR>\n" );
document.write( "Number of divisors: 3\r<BR>\n" );
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document.write( "A handy tool to use is WolframAlpha. <BR>\n" );
document.write( "Type in something like \"divisors of 289\" and it will give the correct list in increasing order.<BR>\n" );
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