To determine the possible number of groups and the number of students in each group, we need to find the divisors of 156 that are between 4 and 12.

Step 1: Find all divisors of 156

We start by determining the divisors of 156:

• 156 ÷ 1 = 156
• 156 ÷ 2 = 78
• 156 ÷ 3 = 52
• 156 ÷ 4 = 39
• 156 ÷ 6 = 26
• 156 ÷ 12 = 13

The divisors of 156 are: 1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, and 156.

Step 2: Identify divisors between 4 and 12

From the list above, the divisors between 4 and 12 are:

• 4
• 6
• 12

Step 3: Calculate the number of groups for each divisor

For each of these divisors, we can calculate the number of groups:

• If there are 4 students per group: 156 ÷ 4 = 39 groups
• If there are 6 students per group: 156 ÷ 6 = 26 groups
• If there are 12 students per group: 156 ÷ 12 = 13 groups

Final Answer:

The possible group sizes and corresponding number of groups are:

• 39 groups with 4 students each
• 26 groups with 6 students each
• 13 groups with 12 students each

Would you like more details or have any questions about this?

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Here are 5 related questions to explore:

1. How do you find the divisors of a number systematically?
2. What is the prime factorization of 156?
3. How would the answer change if the maximum group size was 15 instead of 12?
4. How can we use the greatest common divisor (GCD) to solve similar problems?
5. What if the program director wanted to maximize the number of groups?

Tip: When looking for divisors of a number, try dividing by smaller integers first to avoid missing any!