Problem 7.5. If a platoon of soldiers is divided into brigades of 7 people, then 2 people will not fit into any brigade. If the platoon is divided into brigades of 12 people, then again 2 people will not fit into any brigade. What is the minimum number of soldiers that need to be added to the platoon so that it can be completely divided into brigades of 7 people and also into brigades of 12 people?
Answer: 82.

Solution. A platoon can be entirely divided into brigades of 7 and brigades of 12 if and only if the number of people in it is divisible by $7 \cdot 12=84$.

Remove two people from the platoon. Then the remaining people can be divided into brigades both ways, meaning their number is divisible by 84. Return the two removed people; to reach the next number divisible by 84, we need to add 82 more people.