Find the largest number that will divide exactly the product of four consecutive integers.

As the smallest product of any 4 consecutive natural numbers is 24,$$\left(1\times\ 2\times\ 3\times\ 4\right)$$. So we need to find the largest number, which divides $$\left(1\times\ 2\times\ 3\times\ 4\right)$$ And that is 24.

Rest all products of any 4 consecutive natural numbers, which include 2 odds & 2 evens , have $$\left(2^3\times3\right)$$ as their factors.

Hence, 24 is the largest natural number which divides the product of any 4 consecutive natural numbers.

A is correct choice.