Integer Roots

For a cubic to have 3 distinct roots, we must have f'(x)=0 for 2 values of x, say a and b, also f(a).f(b)<0

From which we have -12<a<12....

but mavarick hasn't written anything abt equal or distinct roots --- otherwise my answer is same as yours

actually I don,t have answer for that question.....
so you may take both condition means equal roots and distinct roots.

If you take both the conditions , then a can take any value according to what x is , because you will always get an integer a such that
x^3 - 13x = -a.

graph will solve the question

Put x = 10 , we get 10³ - 130 + a = 0 [ if we say that it has 3 solutions ]

Or a = -870

If we put a = -870 in x³ - 13x + a = 0 , then we get 3 equal

solutions 10 , 10 , 10 .

But by your graph this solution is not possible.