Greatest Common Factors and Least Common Multiples 
$\mathrm{GCF}(a,b)=4$ and $\mathrm{LCM}(a,b)=96$.
  Find all pairs of whole numbers $a$ and $b$ for which both statements are true. 

I have no clue where to even start with this problem. Thank you for any help!
 A: Both $a$ and $b$ must contain 4 as a factor, since 4 is the greatest common factor.  Thus, $a = 4i$ for some $i$ and $b = 4j$ for some $j$.  And, since it's the greatest common factor, $i$ and $j$ contain no common factors.
Now, the least common multiple of $a$ and $b$ is 96, which means 96 contains all the factors of $a$ and all the factors of $b$.  So, what does this tell us about $i$ and $j$?  Can you take it from there?
A: I use $\gcd(a,b)$ for what you call "greatest common factor" (in my experience, it is more generally called the "greatest common divisor"; go figure).
For any positive integers $a$ and $b$, $\gcd(a,b)\mathrm{lcm}(a,b)=ab$. This is not hard to prove, and you may already know it.
If you do already know it, then you have that $ab = 4\times 96 = 384 = 2^7\times 3$. Now, you want to find values of $a$ and $b$ whose product is $2^7\times 3$, but where the largest common factor they have is exactly $2^2=4$. So $a$ will account for $2^2$; $b$ will account for $2^2$; that leaves you with $2^3\times 3$ still to distribute. Now, the $3$ can go into either $a$ or $b$ and that will not affect the gcd; but you have to be careful with the three remaining factors of $2$: they must all go into $a$ or into $b$ (can you see why?). That limits the possibilities rather strongly.
