smc

The total cost of the pencils can be found by $(\text{students}\cdot \text{pencils purchased by each}\cdot \text{price of each pencil})$. Since $1771$ is the product of three sets of values, we can begin with prime factorization, since it gives some insight into the values: $7,11,23$. Since neither $(C)$ nor $(E)$ are any of these factors, they can be eliminated immediately, leaving $(A)$, $(B)$, and $(D)$. Beginning with $(A)7$, we see that the number of pencils purchased by each student must be either $11$ or $23$. However, the problem states that the price of each pencil must exceed the number of pencils purchased, so we can eliminate this. Continuing with $(B)11$, we can conclude that the only case that fulfils the restrictions are that there are $23$ students who each purchased $7$ such pencils, so the answer is $\boxed{11}$. We can apply the same logic to $(E)$ as we applied to $(A)$ if one wants to make doubly sure.