Carlos attends art class every 4 weeks, chess club every 2 weeks, and fencing lessons every 3 weeks. If he attended all three this week, when will he attend all three again?

Hello!
Denote the minimum number of weeks after which Carlos will attend all three classes during one week as N. Our task is to find that N.
The next art classes will be after 4 weeks since now, then after 4+4=8 weeks since now and, in general, after 4*A weeks since now, where A is any natural number.
For chess club the formula is 2*C, and for fencing lessons the formula is 3*F.
And all these events will happen simultaneously after N weeks, i.e. all these numbers, 4A, 2C and 3F, must be equal to N:
4A = 2C = 3F = N.
 
We see that 4, 2 and 3 are divisors of N, and N, in turn, is a multiple of 4, 2 and 3. This means N is a common multiple of 4, 2 and 3, and we are interested in the least common multiple, N=LCM(4,2,3).
Let's find N=LCM(4,2,3) = LCM(2^2, 2^1, 3^1).
It must contain all prime factors from 4, 2 and 3, they are 2 and 3. And 2 must be at the power 2 to make a multiple of 4. The result is  2^2*3=12.
Thus the event in interest will happen after 12 weeks first (and will happen again after each next 12 weeks).