Please explain me what this question is all about:
In the Mathematics department of a college, there are 60 first-year students, 84 second-year students and 108 third-year students. All of these students are to be divided into project groups such that each group has the same number of first-, second-, and third-year students. What is the smallest possible size of each group?
Best Answer
You could divide into project groups where each group had one first-year student, one second-year student and one third-year student. If you did this, you would have 60 groups, and each group would have the same number of first-years, the same number of second-years, and the same number of third-years, which is good. However, you would also have 24 second-year students and 48 third-year students without a group. This is bad, and we don't want that.
You could have one group of 60 first-years, one group of 84 second-years, and one group of 108 third-years. Then no student would be without a group, which is good. However, then the groups wouldn't have the same number of first-years, the same number of second-years, nor the same number of third-years. This is bad, and we don't want that.
You could divide into groups where each group has 30 first-years, 42 second-years and 54 third-years. Then you would have two groups with equal composition, and no students left over. This is perfect. However, those two groups turn out to be pretty large. Can you make the groups smaller, while still making sure that all students get a group, and all groups have the same composition of first, second and third-years?
This is what the question is about.