What is the least number of three digits perfectly divisible by 4, 6, 8 and 12?

Options

A. 104
B. 120
C. 240
D. 984

Answer

To find the least three-digit number that is perfectly divisible by 4, 6, 8, and 12, we can follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
First, we need to find the LCM of the numbers 4, 6, 8, and 12.

- **Prime Factorization**:
- 4 = 2²
- 6 = 2¹ × 3¹
- 8 = 2³
- 12 = 2² × 3¹

- **Determine the highest power of each prime**:
- For 2, the highest power is 2³ (from 8).
- For 3, the highest power is 3¹ (from 6 and 12).

- **Calculate the LCM**:
LCM = 2³ × 3¹ = 8 × 3 = 24.

### Step 2: Find the Least Three-Digit Number
Now, we need to find the smallest three-digit number that is divisible by 24.

- The smallest three-digit number is 100.
- To find the least three-digit number divisible by 24, we can divide 100 by 24 and round up to the nearest whole number:

\( \frac{100}{24} \approx 4.1667 \)

- Rounding up gives us 5.

### Step 3: Multiply to Find the Number
Now, multiply 5 by 24 to find the least three-digit number:

- \( 5 × 24 = 120 \)

### Conclusion
The least three-digit number perfectly divisible by 4, 6, 8, and 12 is **120**.

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