André's birthday party has fewer than 120 guests. For dinner, he can divide the guests into complete tables of six people or into complete tables of seven people. In both cases, more than 10 tables are needed and all guests are seated at some table. How many guests are there?
We can distribute the total number of guests at tables of 6 or 7, and the number of guests is a multiple of 6 and 7. Since the least common multiple of 6 and 7 is 42, we can have $42, 84, 126, \ldots$ guests. As there are fewer than 120 guests, we can only have 42 or 84 guests. On the other hand, since more than 10 tables are needed, we have more than 60 guests. Therefore, we discard 42, and the number of guests can only be 84.