Kerry invited 23 friends to his pool party. They played a game where everyone had to separate into groups. Each group had the same number of children. The game could not be played with all 24 children in one group, and each group had to have more than 2 children.

Question

Which of the following are ways that they could divide into groups? Select all that apply.

(1 point)
Responses

They can form 4 groups of 6.

They can form 4 groups of 6.

They can form 12 groups of 2.

They can form 12 groups of 2.

They can form 8 groups of 3.

They can form 8 groups of 3.

They can form 4 groups of 9.

They can form 4 groups of 9.

They can form 2 groups of 12.

Answer 1

To find the valid ways to divide Kerry and his 23 friends (24 children total) into groups, we need to consider the following constraints:

1. The number of children in each group must be the same.
2. The game cannot be played with all 24 children in one group (i.e., one group of 24 is not allowed).
3. Each group must contain more than 2 children.

We will look for the divisors of 24 that satisfy these conditions (divisibility > 2 and < 24):

The divisors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24

Considering the constraints:

• A group size of 1 or 2 is not allowed.
• A group size of 24 is not allowed.

Thus, the valid group sizes are: 3, 4, 6, 8, and 12.

Now let's evaluate each option:

1. They can form 4 groups of 6.

   • Yes, \( 6 \times 4 = 24 \) (Valid)

2. They can form 12 groups of 2.

   • No, \( 2 \times 12 = 24 \) but each group needs more than 2 children. (Not valid)

3. They can form 8 groups of 3.

   • Yes, \( 3 \times 8 = 24 \) (Valid)

4. They can form 4 groups of 9.

   • No, \( 4 \times 9 = 36 \) (Not valid)

5. They can form 2 groups of 12.

   • Yes, \( 12 \times 2 = 24 \) (Valid)

Thus, the valid responses are:

• They can form 4 groups of 6.
• They can form 8 groups of 3.
• They can form 2 groups of 12.