45 students are on a school trip. They need to split into smaller groups of equal size. No student can be in a group alone. Find all of the possible group sizes.

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To determine the possible group sizes for splitting 45 students into smaller, equal-sized groups with no student left alone, we'll follow these steps:

1. **Identify Divisors of 45:**
   - First, find all positive divisors of 45. These are numbers that divide 45 without leaving a remainder.
   - The divisors of 45 are: **1, 3, 5, 9, 15, 45**.

2. **Apply the Constraints:**
   - **Equal Group Sizes:** The group size must be a divisor of 45 to ensure all groups have the same number of students.
   - **No Student Left Alone:** This means each group must have at least 2 students. Therefore, we exclude **1** as a group size.
   - **Smaller Groups:** The group size should be smaller than the total number of students (45). Hence, we also exclude **45** since it would mean having just one group.

3. **Determine Valid Group Sizes:**
   - After applying the constraints, the valid group sizes are: **3, 5, 9, 15**.

4. **Verification:**
   - **3 students per group:** 45 ÷ 3 = 15 groups
   - **5 students per group:** 45 ÷ 5 = 9 groups
   - **9 students per group:** 45 ÷ 9 = 5 groups
   - **15 students per group:** 45 ÷ 15 = 3 groups

All these divisions meet the criteria of equal-sized groups with no student left alone.

**Possible Group Sizes:** 3, 5, 9, and 15 students per group.
Possible group sizes are 3, 5, 9, and 15 students per group.

45 students are on a school trip. They need to split into smaller groups of equal size. No student can be in a group alone. Find all of the possible group sizes.

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