Solution:
Suppose the last student solved $n$ problems, and the total number of problems on the test was $p$. Then the total number of correct solutions written was $7p$ (seven per problem), and also equal to $36 + n$ (the sum of the students' scores), so $p = (36 + n)/7$. The smallest $n \geq 0$ for which this is an integer is $n = 6$. But we also must have $n \leq p$, so $7n \leq 36 + n$, and solving gives $n \leq 6$. Thus $n = 6$ is the answer.