Find out the smallest number of three digits which is completely divisible by 4, 8 and 12.

Options

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

Answer

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

### Step 1: Find the Least Common Multiple (LCM)
To find a number that is divisible by 4, 8, and 12, we first need to calculate the LCM of these numbers.

- **Prime factorization**:
- 4 = 2²
- 8 = 2³
- 12 = 2² × 3

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

So, the LCM = 2³ × 3¹ = 8 × 3 = 24.

### Step 2: Find the smallest three-digit number
The smallest three-digit number is 100. We need to find the smallest three-digit number that is divisible by 24.

### Step 3: Divide 100 by 24
Now, we divide 100 by 24 to see how many times 24 fits into 100:
- 100 ÷ 24 = 4.1667 (approximately)

### Step 4: Multiply the whole number by 24
Now, we take the whole number part (which is 4) and multiply it by 24 to find the largest multiple of 24 that is less than 100:
- 4 × 24 = 96 (which is not a three-digit number)

### Step 5: Increase the whole number by 1
Now, we increase the whole number by 1 to find the next multiple of 24:
- 5 × 24 = 120

### Step 6: Verify
Now we verify if 120 is divisible by 4, 8, and 12:
- 120 ÷ 4 = 30 (divisible)
- 120 ÷ 8 = 15 (divisible)
- 120 ÷ 12 = 10 (divisible)

Since 120 is divisible by all three numbers, we conclude that:

**The smallest three-digit number that is completely divisible by 4, 8, and 12 is 120.**

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