36 students need to be divided into teams of equal size. How many different ways can the students be divided (assuming each team must have more than 1 member)?
18 teams of 2
12 teams of 3
9 teams of 4
6 teams of 6
4 teams of 9
3 teams of 12
2 teams of 18
7 ways :)
7 ways.. I agree!!
To find the number of different ways the 36 students can be divided into teams of equal size, we can consider the factors of the number 36.
Step 1: Find the factors of 36
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
Step 2: Eliminate factors that result in a team size of 1
Since each team must have more than 1 member, we can eliminate the factors 1 and 2 from the list.
Step 3: Divide the remaining factors by 2 to find the number of teams
Now, divide each of the remaining factors (3, 4, 6, 9, 12, 18, and 36) by 2 to get the number of teams.
3 ÷ 2 = 1.5 (not possible as each team must have an integer number of students)
4 ÷ 2 = 2 (possible)
6 ÷ 2 = 3 (possible)
9 ÷ 2 = 4.5 (not possible)
12 ÷ 2 = 6 (possible)
18 ÷ 2 = 9 (possible)
36 ÷ 2 = 18 (possible)
Step 4: Count the possible team sizes
From the above divisions, we find that the possible team sizes are 2, 3, 6, 9, and 18.
Step 5: Calculate the number of different ways to form teams
To calculate the number of ways the students can be divided into teams of equal size, we need to divide the total number of students (36) by each of the possible team sizes and count the number of unique divisions.
For a team size of 2: 36 ÷ 2 = 18 teams possible
For a team size of 3: 36 ÷ 3 = 12 teams possible
For a team size of 6: 36 ÷ 6 = 6 teams possible
For a team size of 9: 36 ÷ 9 = 4 teams possible
For a team size of 18: 36 ÷ 18 = 2 teams possible
Step 6: Add up the possible divisions
Finally, add up the number of unique divisions for each team size:
18 + 12 + 6 + 4 + 2 = 42
So, there are a total of 42 different ways the 36 students can be divided into teams of equal size, assuming each team must have more than 1 member.