I know there are manifolds (with or without boundary) $A$ and $B$ such that $A\times C$ is homeomorphic to $B\times C$ but $A$ is NOT homeomorphic to $B$.

My question is (in the Diffeomorphism Category)

Is there any infinitely many $A_i$, with same dimension of course, which are pairwise non-diffeomorphic, but $A_i\times C$ become all diffeomorphic to each other.

1) What's the answer to the question under the assumption on $A_i$, $C$: Smooth Close manifolds.

2) What if we change diffeomorphic to homeomorphic?

3) Is there any example when we require $C$ to be Torus?