The numbers $x_1 = 2, x_2 = 3, x_3 = 4$ belong to the roots of particular polynominal W(x). In addition, all of its coeffitients are integers. Is this polynominal divisible by 24 for any x?
If the numbers 2, 3 and 4 are roots of this polynomial, then it can be written as: $W(x) = a(x-2)(x-3)(x-4)(x-x_4)...(x-x_n)$ In addition, all of the given roots are integers, and so the coefficient a = 1. Hence: $W(x) = (x-2)(x-3)(x-4)(x-x_4)...(x-x_n)$ But how can I prove that this expression is divisible by 24 for any x?
There is some confusion. If $W(x)$ has integer coefficients, it does not mean that all its roots are integers. In general what you can say is that $W(x) = p(x)(x-2)(x-3)(x-4)$ for some polynomial $p(x)$ with integral coefficients.
Now, the dumbest such polynomial is exactly $f(x)=(x-2)(x-3)(x-4)$, and you can see that this is not divisible by $24$ for $x=1$.