How many numbers are there from 300 to 700 which are divisible by 2, 3 and 7?

Options

A. 7
B. 8
C. 9
D. 10

Answer

To find how many numbers between 300 and 700 are divisible by 2, 3, and 7, we can follow these steps:

### Step 1: Determine the range
We need to find the numbers between 300 and 700. The range of interest is from 301 to 699 (inclusive).

### Step 2: Find the least common multiple (LCM)
To find numbers divisible by 2, 3, and 7, we first need to find the LCM of these numbers.

- The prime factorization of the numbers is:
- 2 = 2
- 3 = 3
- 7 = 7

- The LCM is calculated by taking the highest power of each prime:
- LCM(2, 3, 7) = 2^1 * 3^1 * 7^1 = 42

### Step 3: Identify the first and last multiples of 42 within the range
Next, we will find the smallest multiple of 42 that is greater than or equal to 301 and the largest multiple of 42 that is less than or equal to 699.

- **Finding the first multiple of 42 greater than or equal to 301:**
- Divide 301 by 42:
\[ 301 \div 42 \approx 7.1667 \]
- The smallest integer greater than 7.1667 is 8.
- Therefore, the first multiple is:
\[ 42 \times 8 = 336 \]

- **Finding the last multiple of 42 less than or equal to 699:**
- Divide 699 by 42:
\[ 699 \div 42 \approx 16.6429 \]
- The largest integer less than 16.6429 is 16.
- Therefore, the last multiple is:
\[ 42 \times 16 = 672 \]

### Step 4: Count the multiples of 42 from 336 to 672
Now we need to find how many multiples of 42 are there from 336 to 672.

- The multiples of 42 can be expressed as:
\[ 42 \times n \quad \text{where } n \text{ is an integer} \]
- For the range:
- The first multiple corresponds to \( n = 8 \) (336).
- The last multiple corresponds to \( n = 16 \) (672).

### Step 5: Calculate the total count of multiples
To find the total count of integers \( n \) from 8 to 16, we use:
\[ \text{Count} = \text{Last } n - \text{First } n + 1 = 16 - 8 + 1 = 9 \]

### Conclusion
Thus, there are **9 numbers** between 300 and 700 that are divisible by 2, 3, and 7.

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