In the lunch room, 36 fith-graders and 27 fourth-graders are sitting in equal groups. All the students in each group are in the same grade.What is the greatest number of students who could be in each group?

The fifth-graders could evenly be in

1 group of 36 ... or
2 groups of 18 ...or
3 groups of 12 ... or
4 groups of 9 ... or
6 groups of 6 ... or
9 groups of 4 ...or
12 groups of 3 ... or
18 groups of 2 ... or
36 groups of 1

The fourth-graders could evenly be in
1 group of 27...or
3 groups of 9...or
9 groups of 3...or
27 groups of 1
If they have to have the same number in both groups, they could either have 1, 3 or 9 (the common factors). Since we want the greatest number, which would we choose?

To find the greatest number of students who could be in each group, we need to find the largest common factor of 36 and 27.

Step 1: Find the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
Step 2: Find the factors of 27: 1, 3, 9, 27.

Step 3: Determine the common factors: Both 36 and 27 have the factors 1, 3, and 9 in common.

Step 4: Identify the greatest common factor: The largest common factor is 9.

Therefore, the greatest number of students who could be in each group is 9.