I got another math challenge for you guys!

GFC? You mean GCF? As in, Greatest Common Factor?

24 & 36.
I can’t claim a methodical solution, however, just played around with the factors.
72 = 2*2*2*3*3
Factors of 72 are 2,4,6,8,9,12,18,24,36. So the numbers must be in that list.
Since GCF is 12 they have to be multiples of 12, which leaves 12, 24, and 36. Of these, the pairs that differ by 12 are (12,24) and (24,36). The former pair has LCM of 24, which leaves (24,36).

I don’t know how @gasman got this, but you can get the answer algebraically.
If the two numbers are x and y then xy = LCM x GCF = 24×36 = 864
y – x = 12, y = x + 12
Substituting x + 12 for y gives
x(x+12) = 864
That becomes a quadratic equation,.. With a bit of work you could eventually factor it, but it is easier to use the quadratic equation formula.

Here is a better approach. It takes a little more thought, but the calculation is trivial.
The two numbers have the form 12a and 12b, where a and b have no factors in common.
LCM = 12ab = 72, so ab = 6. At this point the solution should be fairly obvious. There are only two possibilities for a and b. To make things even more obvious, use the fact that the difference between the two numbers is 12. That gives 12a – 12b = 12, so a – b =1.

One final note. Using the second technique allows you to work with larger numbers and still have an easy problem to solve. Leave everything the same in your problem, but make the LCM = 360.

@LostInParadise ”If the two numbers are x and y then xy = LCM x GCF.” That’s a good thing to know. It wasn’t obvious to me but I worked out a derivation. Your other approach is elegant and elemental. If 12ab=360 then ab=30 while a=b-1 as before, yielding a=5, b=6. Multiply by 12 to make 60 and 72. Nice.