Let `A = {n in [100, 700] cap N : n` is neither a multiple of 3 nor a multiple of 4}. Then the number of elements in A is

Options

A. 290
B. 300
C. 280
D. 310

Answer

To solve the problem, we need to find the number of integers \( n \) in the range from 100 to 700 that are neither multiples of 3 nor multiples of 4. We will use the principle of inclusion-exclusion to find the solution.

### Step 1: Find the total numbers in the range [100, 700]
The total numbers from 100 to 700 can be calculated as:
\[ \text{Total numbers} = 700 - 100 + 1 = 601 \]

### Step 2: Find the numbers divisible by 3
To find the numbers divisible by 3 in the range, we first identify the smallest and largest multiples of 3 within the range.

- The smallest multiple of 3 greater than or equal to 100 is:
\[ 102 \quad (\text{since } 100 \div 3 = 33.33 \Rightarrow 34 \times 3 = 102) \]

- The largest multiple of 3 less than or equal to 700 is:
\[ 699 \quad (\text{since } 700 \div 3 = 233.33 \Rightarrow 233 \times 3 = 699) \]

Now, we can find how many multiples of 3 are there from 102 to 699. This forms an arithmetic sequence where:
- First term \( a = 102 \)
- Common difference \( d = 3 \)
- Last term \( l = 699 \)

The number of terms \( n \) in this sequence can be found using the formula for the nth term of an arithmetic sequence:
\[ l = a + (n-1)d \]
Substituting the known values:
\[ 699 = 102 + (n-1) \cdot 3 \]
\[ 699 - 102 = (n-1) \cdot 3 \]
\[ 597 = (n-1) \cdot 3 \]
\[ n-1 = 199 \Rightarrow n = 200 \]

So, there are 200 numbers divisible by 3.

### Step 3: Find the numbers divisible by 4
Similarly, we find the numbers divisible by 4 in the range.

- The smallest multiple of 4 greater than or equal to 100 is:
\[ 100 \]

- The largest multiple of 4 less than or equal to 700 is:
\[ 700 \]

Again, we form an arithmetic sequence:
- First term \( a = 100 \)
- Common difference \( d = 4 \)
- Last term \( l = 700 \)

Using the nth term formula:
\[ 700 = 100 + (n-1) \cdot 4 \]
\[ 700 - 100 = (n-1) \cdot 4 \]
\[ 600 = (n-1) \cdot 4 \]
\[ n-1 = 150 \Rightarrow n = 151 \]

So, there are 151 numbers divisible by 4.

### Step 4: Find the numbers divisible by both 3 and 4 (i.e., divisible by 12)
Now, we need to find the numbers that are divisible by both 3 and 4, which means they are divisible by 12.

- The smallest multiple of 12 greater than or equal to 100 is:
\[ 108 \quad (\text{since } 100 \div 12 = 8.33 \Rightarrow 9 \times 12 = 108) \]

- The largest multiple of 12 less than or equal to 700 is:
\[ 696 \quad (\text{since } 700 \div 12 = 58.33 \Rightarrow 58 \times 12 = 696) \]

Again, we form an arithmetic sequence:
- First term \( a = 108 \)
- Common difference \( d = 12 \)
- Last term \( l = 696 \)

Using the nth term formula:
\[ 696 = 108 + (n-1) \cdot 12 \]
\[ 696 - 108 = (n-1) \cdot 12 \]
\[ 588 = (n-1) \cdot 12 \]
\[ n-1 = 49 \Rightarrow n = 50 \]

So, there are 50 numbers divisible by both 3 and 4.

### Step 5: Use Inclusion-Exclusion Principle
Now we can use the inclusion-exclusion principle to find the total number of integers that are divisible by either 3 or 4:
\[ \text{Numbers divisible by 3 or 4} = (\text{Numbers divisible by 3}) + (\text{Numbers divisible by 4}) - (\text{Numbers divisible by both 3 and 4}) \]
\[ = 200 + 151 - 50 = 301 \]

### Step 6: Find the numbers that are neither divisible by 3 nor 4
Finally, we subtract the count of numbers that are divisible by 3 or 4 from the total count of numbers in the range:
\[ \text{Numbers neither divisible by 3 nor 4} = \text{Total numbers} - \text{Numbers divisible by 3 or 4} \]
\[ = 601 - 301 = 300 \]

### Final Answer
The number of elements in set \( A \) is \( \boxed{300} \).