How many numbers between 300 and 785 are exactly divisible by 13?

Options

A. 39
B. 40
C. 41
D. None of the above

Answer

To find how many numbers between 300 and 785 are exactly divisible by 13, we can follow these steps:

### Step 1: Find the first number greater than 300 that is divisible by 13.
To find the first number greater than 300 that is divisible by 13, we can divide 300 by 13 and round up to the nearest whole number.

\[ \text{First number} = 13 \times \lceil \frac{300}{13} \rceil \]

Calculating \( \frac{300}{13} \):

\[ \frac{300}{13} \approx 23.0769 \]

Rounding up gives us 24. Now, we multiply by 13:

\[ 13 \times 24 = 312 \]

So, the first number greater than 300 that is divisible by 13 is **312**.

### Step 2: Find the last number less than 785 that is divisible by 13.
To find the last number less than 785 that is divisible by 13, we can divide 785 by 13 and round down to the nearest whole number.

\[ \text{Last number} = 13 \times \lfloor \frac{785}{13} \rfloor \]

Calculating \( \frac{785}{13} \):

\[ \frac{785}{13} \approx 60.3846 \]

Rounding down gives us 60. Now, we multiply by 13:

\[ 13 \times 60 = 780 \]

So, the last number less than 785 that is divisible by 13 is **780**.

### Step 3: Find the total count of numbers divisible by 13 between 312 and 780.
The numbers divisible by 13 between 312 and 780 form an arithmetic sequence where:
- The first term \( a = 312 \)
- The last term \( l = 780 \)
- The common difference \( d = 13 \)

To find the number of terms \( n \) in this sequence, we can use the formula for the nth term of an arithmetic sequence:

\[ l = a + (n - 1) \cdot d \]

Substituting the known values:

\[ 780 = 312 + (n - 1) \cdot 13 \]

Rearranging gives:

\[ 780 - 312 = (n - 1) \cdot 13 \]

Calculating \( 780 - 312 \):

\[ 468 = (n - 1) \cdot 13 \]

Now, divide both sides by 13:

\[ n - 1 = \frac{468}{13} \]

Calculating \( \frac{468}{13} \):

\[ n - 1 = 36 \]

Adding 1 to both sides gives:

\[ n = 37 \]

### Conclusion
Thus, the total count of numbers between 300 and 785 that are exactly divisible by 13 is **37**.

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