The number nearest to 10000, which is exactly divisible by each of 3, 4, 5, 6, 7 and 8 , is

Options

A. 9240
B. 10080
C. 9996
D. 100000

Answer

To find the number nearest to 10,000 that is exactly divisible by each of 3, 4, 5, 6, 7, and 8, we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
First, we need to find the LCM of the numbers 3, 4, 5, 6, 7, and 8.

- The prime factorization of each number is:
- 3 = 3
- 4 = 2^2
- 5 = 5
- 6 = 2 × 3
- 7 = 7
- 8 = 2^3

- To find the LCM, we take the highest power of each prime number:
- For 2: the highest power is 2^3 (from 8)
- For 3: the highest power is 3^1 (from 3 or 6)
- For 5: the highest power is 5^1 (from 5)
- For 7: the highest power is 7^1 (from 7)

Thus, the LCM is:
\[ \text{LCM} = 2^3 \times 3^1 \times 5^1 \times 7^1 = 8 \times 3 \times 5 \times 7 \]

Calculating this step-by-step:
- \( 8 \times 3 = 24 \)
- \( 24 \times 5 = 120 \)
- \( 120 \times 7 = 840 \)

So, the LCM of 3, 4, 5, 6, 7, and 8 is **840**.

### Step 2: Find the nearest multiples of 840 to 10,000
Next, we need to find the multiples of 840 that are closest to 10,000.

- Divide 10,000 by 840:
\[ \frac{10000}{840} \approx 11.90 \]

- The nearest whole numbers are 11 and 12.
- Now, calculate the multiples:
- For 11:
\[ 11 \times 840 = 9240 \]
- For 12:
\[ 12 \times 840 = 10080 \]

### Step 3: Determine which multiple is closer to 10,000
Now, we compare the distances of these multiples from 10,000:
- Distance from 10,000 to 9240:
\[ 10000 - 9240 = 760 \]
- Distance from 10,000 to 10080:
\[ 10080 - 10000 = 80 \]

Since 80 is less than 760, **10080** is closer to 10,000.

### Final Answer
The number nearest to 10,000 that is exactly divisible by 3, 4, 5, 6, 7, and 8 is **10,080**.

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