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?
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).