How many numbers are there from 200 to 285 which are divisible by 7?

Options

A. 11
B. 12
C. 22
D. 32

Answer

To find how many numbers from 200 to 285 are divisible by 7, we can follow these steps:

### Step 1: Identify the range
We need to find numbers between 200 and 285, inclusive.

### Step 2: Find the smallest number in the range that is divisible by 7
To find the smallest number greater than or equal to 200 that is divisible by 7, we can divide 200 by 7 and round up to the nearest whole number, then multiply by 7.

\[ \text{Smallest number} = \lceil \frac{200}{7} \rceil \times 7 \]

Calculating \( \frac{200}{7} \):

\[ \frac{200}{7} \approx 28.57 \]

Rounding up gives us 29. Now, multiplying by 7:

\[ 29 \times 7 = 203 \]

So, the smallest number in the range that is divisible by 7 is **203**.

### Step 3: Find the largest number in the range that is divisible by 7
To find the largest number less than or equal to 285 that is divisible by 7, we can divide 285 by 7 and round down to the nearest whole number, then multiply by 7.

\[ \text{Largest number} = \lfloor \frac{285}{7} \rfloor \times 7 \]

Calculating \( \frac{285}{7} \):

\[ \frac{285}{7} \approx 40.71 \]

Rounding down gives us 40. Now, multiplying by 7:

\[ 40 \times 7 = 280 \]

So, the largest number in the range that is divisible by 7 is **280**.

### Step 4: Count the numbers divisible by 7 between 203 and 280
Now we need to find how many multiples of 7 are there from 203 to 280. The multiples of 7 can be expressed as:

\[ 7n \quad \text{where } n \text{ is an integer} \]

We already found that:

- The smallest \( n \) for 203 is 29 (since \( 203 = 7 \times 29 \)).
- The largest \( n \) for 280 is 40 (since \( 280 = 7 \times 40 \)).

### Step 5: Calculate the total count
To find the total count of integers \( n \) from 29 to 40, we use:

\[ \text{Count} = 40 - 29 + 1 = 12 \]

Thus, there are **12 numbers** from 200 to 285 that are divisible by 7.

### Final Answer
The answer is **12**.

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