What is the smallest number of 5 digits which is exactly divisible by 12, 24, 48, 60 and 96 ?
A. 10000
B. 10024
C. 10160
D. 10080

Options

A. C
B. A
C. D
D. B

Answer

To find the smallest 5-digit number that is exactly divisible by 12, 24, 48, 60, and 96, we will follow these steps:

### Step 1: Find the LCM of the given numbers
To determine the smallest number that is divisible by all the given numbers, we first need to calculate the Least Common Multiple (LCM) of 12, 24, 48, 60, and 96.

- **Prime factorization:**
- 12 = 2^2 × 3^1
- 24 = 2^3 × 3^1
- 48 = 2^4 × 3^1
- 60 = 2^2 × 3^1 × 5^1
- 96 = 2^5 × 3^1

- **Taking the highest power of each prime:**
- For 2: max(2^2, 2^3, 2^4, 2^2, 2^5) = 2^5
- For 3: max(3^1, 3^1, 3^1, 3^1, 3^1) = 3^1
- For 5: max(5^0, 5^0, 5^0, 5^1, 5^0) = 5^1

- **Calculating the LCM:**
LCM = 2^5 × 3^1 × 5^1 = 32 × 3 × 5 = 480

### Step 2: Identify the smallest 5-digit number
The smallest 5-digit number is 10,000.

### Step 3: Determine the smallest 5-digit number divisible by the LCM
Now, we need to find the smallest 5-digit number that is divisible by 480. We can do this by dividing 10,000 by 480 and rounding up to the nearest whole number.

- **Calculating the division:**
10,000 ÷ 480 ≈ 20.8333

### Step 4: Round up to the nearest whole number
We round up 20.8333 to 21. This means we need to multiply 480 by 21 to find the smallest 5-digit number that is divisible by 480.

### Step 5: Calculate the product
Now, we calculate:
480 × 21 = 10,080

### Conclusion
The smallest 5-digit number that is exactly divisible by 12, 24, 48, 60, and 96 is **10,080**.

### Answer
D. 10080

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