Our general nature is to try to simplify / generalize things by finding patterns. However, although I could intuitively try to apply the idea by extending it to the general case of n variables, in this case it does not work. I will try to break down the argument by the formula.
First we need to understand how the formula LCM (x, y) * GCD (x, y) = x * y could come. To find an LCM or GCD, one way is to break down each of the numbers into their main factors. Let x = 84, y = 30
x = 2 * 2 * 3 * 7 = (2 * 3) * 2 * 7
y = 2 * 3 * 5 = (2 * 3) * 5
The part in parenthesis is the general part. Therefore, we say that ok, 2 * 3, i.e. 6, should be able to divide both x and y and call it a BIG common factor. Keep in mind that only 2 or only 3 are also common divisors of x and y, but not BIG common divisors.
To find the common multiple LESS, we take the GCD and multiply it by all the remaining numbers. So, our LCM is (2 * 3) * 2 * 7 * 5 = 420. I do not describe intuition because it is simple and also not directly related.
So, if you multiply x and y, you get 84 * 30 = (2 * 3 * 2 * 7) * (2 * 3 * 5) = (2 * 3) * ((2 * 3) * 2 * 7 * 5) = GCD (x, y) * LCM (x, y) = (2 * 3) * (2 * 3) * (2 * 7 * 5) = [the general part is raised to degree 2, since it is repeated in both numbers] * [the rest of the left is above the factors in all numbers].
Now, going to your question, if you take another variable z = 18 = (2 * 3) * 3, then the GCD of all three numbers is a common part (2 * 3), i.e. 6 and LCM is (2 * 3) * 2 * 7 * 5 * 3, whatever that is.
Now x * y * z = (2 * 3) * (2 * 3) * (2 * 3) * (2 * 7 * 5 * 3) = [the general part is raised to degree 3, since it repeats in all 3 numbers] * [the rest of the left coefficients in all numbers]. But if you have several GCDs and LCMs, you only get (2 * 3) * ((2 * 3) * 2 * 7 * 5 * 3) = (2 * 3) * (2 * 3) * (2 * 7 * 5 * 3), i.e. The general part is counted only twice, and not thrice.
However, this may also be the case when some of the factors between some numbers (not all, i.e. cannot be included in the GCD) are common. In the general case, n variables GDD (x [1], x [2], ... x [n]) = c [1] c [2] .. c [k], where each of c [i] 1 < = i <= k exists once in all numbers. LCM ((x [1], x [2], ... x [n]) = GCD (x [1], x [2], ... x [n]) * ((h (p [1 ]) * h (p [2]) * ... h (p [l])), where each p [j], 1 <= j <= l is a prime number in the list of left factors that are not part of the GCD and h (p [i]) is the highest degree p [i] present in any of them.
Now, when we multiply LCM and GCD by n numbers, in addition to losing more than twice on GCD factors, we also lose on factors that are partially distributed between some numbers and can only find the result from the higher degrees that are present, multiplies on the gcd.