Which of the following numbers greater than 1000 and also nearest to 10000 is exactly divisible by 21,25 and 30 ?

Options

A. 10910
B. 10800
C. 10500
D. 10710

Answer

To solve the problem of finding a number greater than 1000 and nearest to 10000 that is exactly divisible by 21, 25, and 30, we will follow these steps:

### Step 1: Find the LCM of 21, 25, and 30
To determine the least common multiple (LCM), we first find the prime factorization of each number:
- **21** = 3 × 7
- **25** = 5²
- **30** = 2 × 3 × 5

Now, we take the highest power of each prime number:
- The highest power of **2** is \(2^1\) (from 30)
- The highest power of **3** is \(3^1\) (from 21 and 30)
- The highest power of **5** is \(5^2\) (from 25)
- The highest power of **7** is \(7^1\) (from 21)

Now, we can calculate the LCM:
\[ \text{LCM} = 2^1 \times 3^1 \times 5^2 \times 7^1 = 2 \times 3 \times 25 \times 7 \]
Calculating this step by step:
- \(2 \times 3 = 6\)
- \(6 \times 25 = 150\)
- \(150 \times 7 = 1050\)

Thus, the LCM of 21, 25, and 30 is **1050**.

### Step 2: Find multiples of 1050 that are greater than 1000 and nearest to 10000
Now we need to find multiples of 1050 that are greater than 1000 and closest to 10000.

1. **First multiple greater than 1000**:
- \(1050 \times 1 = 1050\) (This is greater than 1000)

2. **Next multiples**:
- \(1050 \times 2 = 2100\)
- \(1050 \times 3 = 3150\)
- \(1050 \times 4 = 4200\)
- \(1050 \times 5 = 5250\)
- \(1050 \times 6 = 6300\)
- \(1050 \times 7 = 7350\)
- \(1050 \times 8 = 8400\)
- \(1050 \times 9 = 9450\)
- \(1050 \times 10 = 10500\)

### Step 3: Identify the nearest multiple to 10000
From the multiples calculated:
- The multiple \(10500\) is greater than 10000.
- The multiple \(9450\) is less than 10000.

Now we compare the distances:
- Distance from 10000 to 9450 = \(10000 - 9450 = 550\)
- Distance from 10000 to 10500 = \(10500 - 10000 = 500\)

Since \(500 < 550\), the nearest multiple of 1050 to 10000 is **10500**.

### Conclusion
The number that is greater than 1000 and nearest to 10000 that is exactly divisible by 21, 25, and 30 is **10500**.