let the four consecutive numbers be

x-1, x, x+1 and x+2

if one of them is 0 then indeed the product of four consecutive integers is equal to one of them that is 0.

lets assume that none of the four consecutive integers is 0.

let (x-1)x(x+1)(x+2) = (x-1)

x^3 + 3x^2 + 2x - 1 = 0

lets examine the function f(x) = x^3 + 3x^2 + 2x - 1

f(0) = -1 < 0

f(1) = 2 > 0

thus the solution of x^3 + 3x^2 + 2x - 1 = 0 is a number from the interval (0,1) and it is not an integer

click here to see the graph of the function y = x^3 - x - 1

Click to see all the steps

using the same reasoning as above for the cases when

(x-1)x(x+1)(x+2) = x

(x-1)x(x+1)(x+2) = x+1

(x-1)x(x+1)(x+2) = x+2

we conclude that none of this equations has an integer solution.

The largest possible and the only value for one of the integers is 0.