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 December 1st, 2010, 01:09 PM #1 Newbie   Joined: Oct 2010 Posts: 10 Thanks: 0 Please help with this math problem including primes Let the letters p, q, and r represent different primes. Then p^2 qr^3 has 24 divisors. So would p^23. Use p, q, and r to describe all whole numbers having exactly the following number of divisors. (a) 2 (b) 3 (c) 4 (d) 5 (e) 6 (f) 12
 December 1st, 2010, 03:26 PM #2 Senior Member   Joined: Nov 2010 From: Staten Island, NY Posts: 152 Thanks: 0 Re: Please help with this math problem including primes (a) $p$ (b) $p^2$ (c) $p^3 , pq$ (d) $p^4$ (e) $p^5, p^2 q$ (f) $p^{11}$ , $p^5 q$ , $p^3 q^2$ , $p^2 qr$ In general, we get these by looking at all factorizations of the number, and using k-1 as an exponent for each factor k. For example, the factorizations of 12 are 12*1, 6*2, 4*3, and 3*2*2
December 1st, 2010, 04:31 PM   #3
Math Team

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Hello, sophieT66!

Quote:
 $\text{Let the letters }p,\,q,\,r\text{ represent different primes.}$ $\text{Then }p^2qr^3\text{ has 24 divisors. }\:\text{So would }p^{23}.$ [color=beige] .[/color] [color=red]**[/color] $\text{Use }p,\, q,\text{ and }r\text{ to describe all whole numbers having exactly the following number of divisors.}$ [color=beige]. . [/color]$(a)\; 2\;\;\;(b)\; 3 \;\;\; (c)\; 4 \;\;\; (d)\; 5 \;\;\;(e)\; 6 \;\;\;(f)\; 12$

[color=red]**[/color]
$\text{If the prime factorization of }N\text{ is: }\;N \:=\:p^a\,\cdot\,q^b\,\cdot\,r^c\,\cdots\,u^n$

[color=beige]. . [/color]$\text{the number of divisors of }N\text{ is: }\:(a\,+\,1)(b\,+\,1)(c\,+\,1)\,\cdots\,(n\,+\,1)$

$\text{(Add one to each exponent and multiply.)}$

$\text{(a) 2 divisors: }\;\;\;\begin{Bmatrix}p & q & r \end{Bmatrix}$

$\text{(b) 3 divisors: }\;\;\;\begin{Bmatrix}p^2 & q^2 & r^2 \end{Bmatrix}$

$\text{(c) 4 divisors: }\;\;\; \begin{Bmatrix} p^3 & q^3 & r^3 \\ \\ \\ pq & qr & pr \end{Bmatrix}$

$\text{(d) 5 divisors: }\;\;\;\begin{Bmatrix}p^4 & q^4 & r^4 \end{Bmatrix}$

$\text{(e) 6 divisors: }\;\;\;\begin{Bmatrix} & p^5 & q^5 & r^5 \\ \\ \\ p^2q & pq^2 & q^2r & qr^2 & p^2r & pr^2 \end{Bmatrix}$

$\text{(f) 12 divisors: }\;\;\begin{Bmatrix} & p^{11} & q^{11} & r^{11} \\ \\ \\ p^5q & pq^5 & q^5r & qr^5 & p^5r & pr^5 \\ \\ \\ p^3q^2 & p^2q^3 & q^3r^2 & q^2r^3 & p^3r^2 & p^2r^3 \\ \\ \\ & p^2qr & pq^2r & pqr^2 \end{Bmatrix}$

 December 2nd, 2010, 12:04 PM #4 Senior Member   Joined: Nov 2010 From: Staten Island, NY Posts: 152 Thanks: 0 Re: Please help with this math problem including primes Good post Soroban, but one small correction: pqr has 8 divisors, not 6.
 December 2nd, 2010, 04:41 PM #5 Math Team   Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408 Re: Please help with this math problem including primes Thanks for the heads-up, Dr.Steve ! Hello, Moderators: We don't have an EDIT capability?
 December 2nd, 2010, 04:55 PM #6 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Please help with this math problem including primes The editing of posts was recently increased from 1 hour to 12 hours after the post is made. I'm sure [color=#BF0000]julien[/color] has a good reason for imposing a time limit on post editing. I edited your post.

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