equations with lcm and gcf If $p$, $q$, and $r$ are three different odd prime numbers what are the $\operatorname{lcm} (2pq, 2pr, 2qr)$ and $\operatorname{gcf} (2pq, 2pr, 2qr)$. Anyone have any suggestions on how to even start this problem? I am just back in college and really struggling with this stuff. Thank you.
 A: Any multiple of $2pq$, $2pr$, and $2qr$ must contain factors of $2, p, q, r$.  Thus, any multiple is at least $2pqr$.  What can you say from there?
As far as the greatest common factor of the three, start by finding one common factor of all three parts.  Are there any others?
Example (since the letters are confusing you):
Let's let $p = 3$, $q = 5$, and $r = 7$?  That way there won't be any letters.  It's just find the lcm and gcf of 30, 42, and 70.
The easiest way to do this example is list all factors of each number:
The factors of 30 are 2, 3, 5, 6, 10, 15, 30
The factors of 42 are 2, 3, 7, 6, 14, 21, 42
The factors of 70 are 2, 5, 7, 10, 14, 35, 70
What is the greatest common factor?  That means, what is the biggest number that appears in each list?  The answer is 2.
Now, for the least common multiple.  We list the multiples of each number as big lists. Then, we look for the smallest number that appears in each list.
Multiples of 30 are 30, 60, 90, 120, 150, 210, 240, ...
Multiples of 42 are 42, 84, 126, 168, 210, 252, ...
Multiples of 70 are 70, 140, 210, 280, 350, ...
What's the smallest number that appears in all lists?  It's 210.  That's the least common multiple.
That just gave us the answer for this one specific case of the problem.  If you do the problem as stated, you'll find the answer that works for any 3 odd primes.
A: *

*$\mathrm GCF(2pr,2qr,2pq)$  
The divisors of "$2pr$" are: $1$, $2$, $p$, $r$, $2p$, $2r$, $pr$ and $2pr$ (while $p$, $q$ and $r$ are odd
primes)
The divisors of "$2pq$" are:    $1$, $2$, $p$, $q$, $2p$, $2q$, $pq$ and $2pq$ (while $p$, $q$ and $r$ are odd
primes)
The divisors of "$2qr$" are: $1$, $2$, $r$, $q$, $2r$, $2q$, $qr$ and $2qr$ (while $p$, $q$ and $r$ are odd
primes)
Hence, what's the greatest common factor of the three?

*$\mathrm LCM(2pq,2pr,2qr)$
Multiples of "$2pr$" are:  $2pr$, $\dots$, $2pqr$, $\dots$  
Multiples of "$2pq$" are:  $2pq$, $\dots$, $2pqr$, $\dots$
Multiples of "$2qr$" are:  $2qr$, $\dots$, $2pqr$, $\dots$
What's the least common multiple of the three?

See also: Greatest Common Divisor and Least Common Multiple
