To solve this question, we need to know two facts.
The product of 4 consecutive numbers is always divisible by 4!.
Since, we have 4 even numbers, we have an additional 2 available with each number.
Now, using both the facts, we can say that the product of 4 consecutive even numbers is always divisible by,
$=(2^4) \times 4! $
$= 16 \times 24$
Edit: Than you Joe for explaining fact 1 in the comments.
Edit 2: For an alternative solution, check comment by Gopal.