Let S={x|x is a positive multiple of 3 less than 100}, P= {x|x is a prime number less than 20}.

Options

A. 34
B. 41
C. 33
D. 30

Answer

To solve the problem, we need to find the number of elements in the sets \( S \) and \( P \) and then add these two quantities together. Let's break it down step by step.

### Step 1: Define the set \( S \)
The set \( S \) is defined as:
\[ S = \{ x \mid x \text{ is a positive multiple of 3 less than 100} \} \]

To find the elements of \( S \), we list the positive multiples of 3 that are less than 100:
- The first positive multiple of 3 is 3.
- The next multiples are 6, 9, 12, ..., up to the largest multiple of 3 that is less than 100.

The largest multiple of 3 less than 100 can be found by dividing 99 by 3:
\[ 99 \div 3 = 33 \]
Thus, the multiples of 3 less than 100 are:
\[ S = \{ 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99 \} \]

### Step 2: Count the elements in set \( S \)
To count the number of elements in \( S \), we can observe that the multiples of 3 form an arithmetic sequence where:
- First term \( a = 3 \)
- Common difference \( d = 3 \)
- Last term \( l = 99 \)

The number of terms \( n \) in an arithmetic sequence can be calculated using the formula:
\[ n = \frac{l - a}{d} + 1 \]
Substituting the values:
\[ n = \frac{99 - 3}{3} + 1 = \frac{96}{3} + 1 = 32 + 1 = 33 \]

So, \( n(S) = 33 \).

### Step 3: Define the set \( P \)
The set \( P \) is defined as:
\[ P = \{ x \mid x \text{ is a prime number less than 20} \} \]

The prime numbers less than 20 are:
\[ P = \{ 2, 3, 5, 7, 11, 13, 17, 19 \} \]

### Step 4: Count the elements in set \( P \)
Counting the elements in \( P \):
- The prime numbers listed are 2, 3, 5, 7, 11, 13, 17, and 19.
- There are a total of 8 prime numbers.

So, \( n(P) = 8 \).

### Step 5: Add the number of elements in sets \( S \) and \( P \)
Now we add the number of elements in both sets:
\[ n(S) + n(P) = 33 + 8 = 41 \]

### Final Answer
The final answer is:
\[ \text{The number of elements in } S + P = 41 \]

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