The Square of My Age

Answer: Thom is $7$ and Lauren is $13$

The square of Thom's age is at most $62$, so Thom is at most $7$. The square of Thom's age is therefore at most $49$, so Lauren is at least $13$.

The square of Lauren's age is at most $176$, so Lauren is at most $13$. Therefore Lauren must be $13$, and Thom must be $7$.
 

And $114 = 2\times57 = 3\times38 = 6\times19