Find the number of numbers lying between 1 and 1000 which are divisible by each of 6,7 and 15.

Answer

To find the number of numbers lying between 1 and 1000 that are divisible by each of 6, 7, and 15, we will follow these steps:

### Step 1: Find the LCM of 6, 7, and 15

1. **Prime Factorization**:
- 6 = 2 × 3
- 7 = 7
- 15 = 3 × 5

2. **Identify the highest powers of each prime**:
- The prime factors we have are 2, 3, 5, and 7.
- The highest power of 2 is \(2^1\) (from 6).
- The highest power of 3 is \(3^1\) (from both 6 and 15).
- The highest power of 5 is \(5^1\) (from 15).
- The highest power of 7 is \(7^1\) (from 7).

3. **Calculate the LCM**:
\[ \text{LCM} = 2^1 \times 3^1 \times 5^1 \times 7^1 = 2 \times 3 \times 5 \times 7 \]
\[ = 2 \times 3 = 6 \]
\[ = 6 \times 5 = 30 \]
\[ = 30 \times 7 = 210 \]

### Step 2: Count the multiples of the LCM between 1 and 1000

1. **Determine the range**:
- We need to find how many multiples of 210 are there between 1 and 1000.

2. **Find the largest multiple of 210 less than or equal to 1000**:
- We can find this by dividing 1000 by 210:
\[ \frac{1000}{210} \approx 4.76 \]
- The largest integer less than or equal to 4.76 is 4.

3. **Calculate the multiples**:
- The multiples of 210 are:
- \(1 \times 210 = 210\)
- \(2 \times 210 = 420\)
- \(3 \times 210 = 630\)
- \(4 \times 210 = 840\)
- The next multiple, \(5 \times 210 = 1050\), exceeds 1000.

### Conclusion

The numbers between 1 and 1000 that are divisible by 6, 7, and 15 are 210, 420, 630, and 840. Thus, there are **4 numbers**.

### Final Answer:
**4**

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