Eleanor is making chocolate chip cookies for her friends. If she divides the cookies equally among $11$ of her friends, she'll have $4$ cookies left over. If she divides the cookies equally among $7$ of her friends, she'll have $1$ cookie left over. Assuming that Eleanor made fewer than $100$ cookies, what is the sum of the possible numbers of cookies that she could have made?
I need to find the numbers of cookies that are multiples of $11$ plus $4$, and also multiples of $7$ plus $1$. One way to do this is to list the multiples of $11$ plus $4$ until I reach $100$, and then check which ones are also multiples of $7$ plus $1$. The multiples of $11$ plus $4$ are $15, 26, 37, 48, 59, 70, 81, 92$. The multiples of $7$ plus $1$ are $8, 15, 22, 29, 36, 43, 50, 57, 64, 71, 78, 85, 92, 99$. The numbers that appear in both lists are $15, 92$. So the possible numbers of cookies that Eleanor could have made are $15$ and $92$. The sum of these numbers is $15 + 92 = 107$. 