A natural number is chosen at random from the first 100 natural numbers. What is the probability that the number chosen is a multiple of 2 or 3 or 5?

Options

A. `30/100`
B. `1/33`
C. `74/100`
D. `7/10`

Answer

To find the probability that a randomly chosen natural number from the first 100 natural numbers is a multiple of 2, 3, or 5, we can use the principle of inclusion-exclusion. Here are the steps to solve the problem:

### Step 1: Define the Sets
Let:
- \( A \) = Set of multiples of 2
- \( B \) = Set of multiples of 3
- \( C \) = Set of multiples of 5

### Step 2: Calculate the Size of Each Set
1. **Multiples of 2**:
\[ |A| = \left\lfloor \frac{100}{2} \right\rfloor = 50 \]

2. **Multiples of 3**:
\[ |B| = \left\lfloor \frac{100}{3} \right\rfloor = 33 \]

3. **Multiples of 5**:
\[ |C| = \left\lfloor \frac{100}{5} \right\rfloor = 20 \]

### Step 3: Calculate the Size of the Intersections
1. **Multiples of both 2 and 3 (i.e., multiples of 6)**:
\[ |A \cap B| = \left\lfloor \frac{100}{6} \right\rfloor = 16 \]

2. **Multiples of both 3 and 5 (i.e., multiples of 15)**:
\[ |B \cap C| = \left\lfloor \frac{100}{15} \right\rfloor = 6 \]

3. **Multiples of both 2 and 5 (i.e., multiples of 10)**:
\[ |A \cap C| = \left\lfloor \frac{100}{10} \right\rfloor = 10 \]

4. **Multiples of 2, 3, and 5 (i.e., multiples of 30)**:
\[ |A \cap B \cap C| = \left\lfloor \frac{100}{30} \right\rfloor = 3 \]

### Step 4: Apply the Inclusion-Exclusion Principle
To find the total number of favorable outcomes (i.e., multiples of 2, 3, or 5), we use the formula:
\[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |A \cap C| + |A \cap B \cap C| \]

Substituting the values we calculated:
\[ |A \cup B \cup C| = 50 + 33 + 20 - 16 - 6 - 10 + 3 \]
\[ |A \cup B \cup C| = 103 - 32 = 71 \]

### Step 5: Calculate the Probability
The probability \( P \) that a randomly chosen number is a multiple of 2, 3, or 5 is given by:
\[ P = \frac{|A \cup B \cup C|}{100} = \frac{71}{100} \]

### Final Answer
Thus, the probability that the number chosen is a multiple of 2, 3, or 5 is:
\[ \frac{71}{100} \]

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