To find the smallest 3-digit number that is exactly divisible by 6, 8, and 12, we first need to determine the least common multiple (LCM) of these numbers. Here’s how to approach the problem step-by-step.

Understanding the Numbers
- **Divisibility Rules**:
- A number is divisible by **6** if it is divisible by both **2** and **3**.
- A number is divisible by **8** if it ends in three zeros or is of the form 2^3.
- A number is divisible by **12** if it is divisible by both **3** and **4**.

Finding the LCM
- **Prime Factorization**:
- 6 = 2 × 3
- 8 = 2^3
- 12 = 2^2 × 3
- **Taking the Highest Powers**:
- For 2: Highest power is \(2^3\) from 8.
- For 3: Highest power is \(3^1\) from 6 and 12.
- **LCM Calculation**:
- LCM = \(2^3 × 3^1 = 8 × 3 = 24\)

Finding the Smallest 3-Digit Number
- **Identifying the Range**:
- The smallest 3-digit number is **100**.
- **Finding Divisible Numbers**:
- Now, we need to find the smallest 3-digit number divisible by 24.
- **Calculation**:
- Divide 100 by 24: \(100 ÷ 24 ≈ 4.1667\).
- Multiply the next whole number (5) by 24: \(5 × 24 = 120\).

Conclusion
- **Result**:
- The smallest 3-digit number that is exactly divisible by 6, 8, and 12 is **120**.