One method for this question is to notice that the square of Thomas' age is at most $62$, so Thomas is at most $7$. The square of Thomas' 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 Thomas must be $7$.
This problem is taken from the UKMT Mathematical Challenges.