Consecutive numbers say 3*4*5 are divisible by the number of terms in the series i.e.3 in this case.

Take 21*22*23*24 is divisible by 4.

yes.

every third integer is a multiple of 3:
0 1 2 3 4 5 6 7 8 9 ... etc

so if you have the product of three consecutive integers, then one of them has to be a multiple of 3 (colored in red).

furthermore, as another poster has pointed out, this is the case not just for 3, but for an arbitrary # of consecutive integers.

Ok. That confirms my understanding.

Excellent! Glad it helped!