It is very simple:
1. Every second consecutive integer is divided by 2.
2. Every third consecutive integer is divided by 3.
3. Every forth consecutive integer is divided by 4=2*2.
So, you see, there will be 1 integer divided by 4, at least 1 integer divided by 3, two integers divided by 2.
As 4 is divided by 2, then there will be 1 integer that will be divided by 2 and will not divided by 4.
So, there will be these known component dividers in result product: (2*2)*2*3=24, thus this product will be divided by 24.
Sorry, my mathematical language is poor - I never used English in mathematics.