How many even integers n, `13 le n le 313` are of the from 3k + 4, where k is any natural number?

Options

A. A)101
B. B)51
C. C)50
D. D)none of these

Answer

To solve the problem of finding how many even integers \( n \) in the range \( 13 \leq n \leq 313 \) are of the form \( 3k + 4 \) where \( k \) is any natural number, we can follow these steps:

### Step 1: Identify the form of \( n \)
We know that \( n \) can be expressed as:
\[ n = 3k + 4 \]
where \( k \) is a natural number (i.e., \( k \in \{1, 2, 3, \ldots\} \)).

### Step 2: Determine the range of \( k \)
We need to find the values of \( k \) such that \( n \) falls within the range \( 13 \leq n \leq 313 \).

1. **Lower Bound**: Set \( n = 13 \):
\[ 3k + 4 \geq 13 \implies 3k \geq 9 \implies k \geq 3 \]

2. **Upper Bound**: Set \( n = 313 \):
\[ 3k + 4 \leq 313 \implies 3k \leq 309 \implies k \leq 103 \]

Thus, \( k \) can take values from \( 3 \) to \( 103 \).

### Step 3: Count the values of \( k \)
The values of \( k \) form an arithmetic sequence starting from \( 3 \) to \( 103 \). The number of terms in this sequence can be calculated as follows:
\[ \text{Number of terms} = \text{Last term} - \text{First term} + 1 = 103 - 3 + 1 = 101 \]

### Step 4: Determine even integers
Next, we need to find how many of these integers \( n \) are even.

From our expression \( n = 3k + 4 \):
- Since \( 4 \) is even, \( n \) will be even if \( 3k \) is even.
- \( 3k \) is even when \( k \) is even (since \( 3 \) is odd, it does not affect the evenness).

### Step 5: Identify even \( k \)
The even values of \( k \) between \( 3 \) and \( 103 \) are:
- The smallest even \( k \) is \( 4 \).
- The largest even \( k \) is \( 102 \).

The even integers from \( 4 \) to \( 102 \) can be calculated:
\[ \text{Even integers} = 4, 6, 8, \ldots, 102 \]
This is also an arithmetic sequence where:
- First term \( a = 4 \)
- Common difference \( d = 2 \)
- Last term \( l = 102 \)

### Step 6: Count the number of even \( k \)
To find the number of even terms:
\[ \text{Number of even terms} = \frac{l - a}{d} + 1 = \frac{102 - 4}{2} + 1 = \frac{98}{2} + 1 = 49 + 1 = 50 \]

### Conclusion
Thus, the total number of even integers \( n \) in the range \( 13 \leq n \leq 313 \) that are of the form \( 3k + 4 \) is \( 50 \).

### Final Answer
The answer is \( 50 \).