gcd(100, 997) = 1. The congruence is solvable and it has unique solution.
100x≡11(mod997)
⟹1000x≡110(mod997)
⟹3x≡110(mod997)
⟹999x≡36630(mod997)
⟹2x≡738(mod997)
⟹998x≡499×738(mod997)
⟹x≡369(mod997)
369 is the required unique 3-digit positive integer.
