Junior Mathematical Olympiad, September 2019

In the puzzle below, $a,b,c,d$, and $e$ are nonzero digits such that the two calculations are correct. The digits need not be distinct. How many solutions are there for which $a<b$?

$ab\times ab=cde\text{and}ba\times ba=edc.$

A) 1 B) 2 C) 3 D) 4 E) 5