How many even integers n, where `100lenle200`, are divisible neither by seven nor by nine ?

Options

A. 40
B. 37
C. 39
D. 38

Answer

To solve the problem of finding how many even integers \( n \) in the range \( 100 \leq n \leq 200 \) are divisible neither by 7 nor by 9, we can follow these steps:

### Step 1: Identify the range of even integers
The even integers between 100 and 200 can be identified as follows:
- The smallest even integer in this range is 100.
- The largest even integer in this range is 200.
- The even integers can be represented as \( 100, 102, 104, \ldots, 200 \).

### Step 2: Count the total even integers in the range
To find the total number of even integers from 100 to 200, we can use the formula for the \( n \)-th term of an arithmetic sequence:
\[ n = a + (k-1)d \]
where:
- \( a \) is the first term (100),
- \( d \) is the common difference (2),
- \( k \) is the number of terms.

Setting the last term equal to 200:
\[ 200 = 100 + (k-1) \cdot 2 \]
Solving for \( k \):
\[ 200 - 100 = (k-1) \cdot 2 \\ 100 = (k-1) \cdot 2 \\ k-1 = 50 \\ k = 51 \]
Thus, there are 51 even integers from 100 to 200.

### Step 3: Count the even integers divisible by 7
Next, we find the even integers in this range that are divisible by 7.
- The smallest even integer divisible by 7 in this range is 105 (7 * 15).
- The largest even integer divisible by 7 in this range is 196 (7 * 28).
- The even integers divisible by 7 can be represented as \( 14, 28, 42, \ldots \).

To find the count:
- The sequence of even integers divisible by 7 is \( 14, 28, 42, \ldots \).
- The first term \( a = 14 \) and the last term \( l = 196 \) with a common difference \( d = 14 \).

Using the formula for the \( n \)-th term:
\[ 196 = 14 + (k-1) \cdot 14 \]
Solving for \( k \):
\[ 196 - 14 = (k-1) \cdot 14 \\ 182 = (k-1) \cdot 14 \\ k-1 = 13 \\ k = 14 \]
Thus, there are 14 even integers between 100 and 200 that are divisible by 7.

### Step 4: Count the even integers divisible by 9
Next, we find the even integers in this range that are divisible by 9.
- The smallest even integer divisible by 9 in this range is 108 (9 * 12).
- The largest even integer divisible by 9 in this range is 198 (9 * 22).

To find the count:
- The sequence of even integers divisible by 9 is \( 108, 126, 144, \ldots \).
- The first term \( a = 108 \) and the last term \( l = 198 \) with a common difference \( d = 18 \).

Using the formula for the \( n \)-th term:
\[ 198 = 108 + (k-1) \cdot 18 \]
Solving for \( k \):
\[ 198 - 108 = (k-1) \cdot 18 \\ 90 = (k-1) \cdot 18 \\ k-1 = 5 \\ k = 6 \]
Thus, there are 6 even integers between 100 and 200 that are divisible by 9.

### Step 5: Count the even integers divisible by both 7 and 9 (i.e., 63)
Next, we find the even integers in this range that are divisible by both 7 and 9.
- The least common multiple of 7 and 9 is 63. The smallest even integer divisible by 63 in this range is 126.
- The largest even integer divisible by 63 in this range is 189.

To find the count:
- The sequence of even integers divisible by 63 is \( 126, 189 \).
- The first term \( a = 126 \) and the last term \( l = 189 \) with a common difference \( d = 126 \).

Using the formula for the \( n \)-th term:
\[ 189 = 126 + (k-1) \cdot 126 \]
Solving for \( k \):
\[ 189 - 126 = (k-1) \cdot 126 \\ 63 = (k-1) \cdot 126 \\ k-1 = 0 \\ k = 1 \]
Thus, there is 1 even integer between 100 and 200 that is divisible by both 7 and 9.

### Step 6: Apply the principle of inclusion-exclusion
Now, we can find the total number of even integers that are divisible by either 7 or 9:
\[ \text{Total divisible by 7 or 9} = (\text{Divisible by 7}) + (\text{Divisible by 9}) - (\text{Divisible by both 7 and 9}) \]
\[ = 14 + 6 - 1 = 19 \]

### Step 7: Calculate the final count
Finally, we subtract the count of even integers that are divisible by either 7 or 9 from the total count of even integers:
\[ \text{Even integers not divisible by 7 or 9} = \text{Total even integers} - \text{Total divisible by 7 or 9} \]
\[ = 51 - 19 = 32 \]

### Final Answer
Thus, the number of even integers \( n \) where \( 100 \leq n \leq 200 \) that are divisible neither by 7 nor by 9 is **32**.