Let $p$ be prime in $[T,2T]$ where $T$ is a parameter.

1. Can we have an integer pair $a,b$ satisfying $ab\equiv c'\bmod p$ such that $c'$ is of size $O(polylog(T))$ and $a,b$ are of size $O(T^{1/2 +\epsilon})$?

2. Given $p$ how many such generically positioned pairs are possible?