# Doubt on integrality gap and LP relaxation

I have an exercise that tells me that, given a problem P (of which now I omit the description) there is no integrality gap between LP and ILP formulation of this problem, and for every fractional LP-solution, there exists an integral feasible solution with the same cost. Then I have to design a poly-time algorithm that from any given optimal LP-solution, computes such an optimal integer assignment.

The fact that there is integrality gap what does it mean in this case? I mean, if I design an approximation-algorithm to compute an optimal integer assignment from a given optimal LP-solution, I will get a solution with object value like something * Optimal value of LP problem . Isn't this in contrast with the fact that there's no IG? It should mean that the optimal value of LP and ILP should be the same, no?