say a,b,c,d are the consecutive numbers. Among them,
there are two even numbers definetly. by the same reasoning,
there is at least one number divisible by 3 and
there is exactly one number divisible by 4.
thus, numbers 2,3 and 4 have their representatives (multiples) contributing to the product.
hence 2*3*4=24 should divide a*b*c*d