What is the smallest 5-digit number which when divided by 8, 14, 24 and 56 leaves zero remainder?

Options

A. 10084
B. 10081
C. 10080
D. 10088

Answer

To find the smallest 5-digit number that is divisible by 8, 14, 24, and 56, we need to follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
To determine the smallest number that is divisible by all four numbers, we first need to find the LCM of 8, 14, 24, and 56.

1. **Prime Factorization**:
- **8** = \(2^3\)
- **14** = \(2^1 \times 7^1\)
- **24** = \(2^3 \times 3^1\)
- **56** = \(2^3 \times 7^1\)

2. **Identify the highest power of each prime factor**:
- For \(2\): The highest power is \(2^3\) (from 8, 24, and 56).
- For \(3\): The highest power is \(3^1\) (from 24).
- For \(7\): The highest power is \(7^1\) (from 14 and 56).

3. **Calculate the LCM**:
\[ \text{LCM} = 2^3 \times 3^1 \times 7^1 = 8 \times 3 \times 7 \]
- First, calculate \(8 \times 3 = 24\).
- Then, calculate \(24 \times 7 = 168\).
- Thus, \(\text{LCM} = 168\).

### Step 2: Find the Smallest 5-Digit Number
Now that we have the LCM, we need to find the smallest 5-digit number that is a multiple of 168.

1. **Determine the smallest 5-digit number**: The smallest 5-digit number is 10000.

2. **Divide 10000 by 168** to find the smallest integer \(n\) such that \(n \times 168\) is a 5-digit number:
\[ n = \frac{10000}{168} \approx 59.52 \]
Since \(n\) must be an integer, we round up to the next whole number, which is 60.

3. **Calculate the smallest 5-digit number**:
\[ 60 \times 168 = 10080 \]

### Conclusion
The smallest 5-digit number which when divided by 8, 14, 24, and 56 leaves zero remainder is **10080**.

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