Find the number of numbers lying between 1 and 1,00,000 which are divisible by each of 15,35 and 77 and are even also .

Answer

To find the number of numbers lying between 1 and 100,000 that are divisible by 15, 35, and 77 and are also even, we can follow these steps:

### Step 1: Find the LCM of 15, 35, and 77

1. **Prime Factorization**:
- \( 15 = 3 \times 5 \)
- \( 35 = 5 \times 7 \)
- \( 77 = 7 \times 11 \)

2. **LCM Calculation**:
- Take the highest power of each prime factor:
- \( 3^1 \) from 15
- \( 5^1 \) from both 15 and 35
- \( 7^1 \) from both 35 and 77
- \( 11^1 \) from 77
- Therefore, the LCM is:
\[ LCM = 3^1 \times 5^1 \times 7^1 \times 11^1 = 3 \times 5 \times 7 \times 11 = 1155 \]

### Step 2: Ensure the number is even

Since 1155 is odd, we need to multiply it by an even number to ensure the result is even. The smallest even number we can multiply by is 2.

\[ \text{Even LCM} = 2 \times 1155 = 2310 \]

### Step 3: Find the range of multiples of 2310 between 1 and 100,000

To find how many multiples of 2310 lie between 1 and 100,000, we can use the formula for the number of multiples of a number \( n \) up to \( N \):
\[ \text{Number of multiples} = \left\lfloor \frac{N}{n} \right\rfloor \]

In this case, \( N = 100,000 \) and \( n = 2310 \):
\[ \text{Number of multiples} = \left\lfloor \frac{100000}{2310} \right\rfloor \]

Calculating this:
\[ \frac{100000}{2310} \approx 43.29 \]
Taking the floor function:
\[ \left\lfloor 43.29 \right\rfloor = 43 \]

### Final Answer

Thus, the number of numbers lying between 1 and 100,000 that are divisible by 15, 35, and 77 and are even is **43**.

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