Find the number of positive intergers up to 100 which are not divisible by any 2,3 and 5 ?

Options

A. `24`
B. `25`
C. `21`
D. `27`

Answer

To find the number of positive integers up to 100 that are not divisible by 2, 3, or 5, we can use the principle of inclusion-exclusion. Here’s a step-by-step solution:

### Step 1: Count the total numbers divisible by each number
1. **Divisible by 2**:
- The numbers divisible by 2 up to 100 are: 2, 4, 6, ..., 100.
- The count is \( \frac{100}{2} = 50 \).

2. **Divisible by 3**:
- The numbers divisible by 3 up to 100 are: 3, 6, 9, ..., 99.
- The count is \( \frac{100}{3} = 33 \) (we take the integer part).

3. **Divisible by 5**:
- The numbers divisible by 5 up to 100 are: 5, 10, 15, ..., 100.
- The count is \( \frac{100}{5} = 20 \).

### Step 2: Count the numbers divisible by pairs of numbers
1. **Divisible by both 2 and 3 (LCM = 6)**:
- The count is \( \frac{100}{6} = 16 \).

2. **Divisible by both 3 and 5 (LCM = 15)**:
- The count is \( \frac{100}{15} = 6 \).

3. **Divisible by both 2 and 5 (LCM = 10)**:
- The count is \( \frac{100}{10} = 10 \).

### Step 3: Count the numbers divisible by all three numbers
1. **Divisible by 2, 3, and 5 (LCM = 30)**:
- The count is \( \frac{100}{30} = 3 \).

### Step 4: Apply the principle of inclusion-exclusion
Using the counts from the previous steps, we can find the total number of integers up to 100 that are divisible by at least one of 2, 3, or 5:

\[ N(2 \cup 3 \cup 5) = N(2) + N(3) + N(5) - N(2 \cap 3) - N(3 \cap 5) - N(2 \cap 5) + N(2 \cap 3 \cap 5) \]

Substituting the values we calculated:

\[ N(2 \cup 3 \cup 5) = 50 + 33 + 20 - 16 - 6 - 10 + 3 \]

Calculating this gives:

\[ N(2 \cup 3 \cup 5) = 50 + 33 + 20 - 16 - 6 - 10 + 3 = 74 \]

### Step 5: Calculate the numbers not divisible by 2, 3, or 5
Now, to find the numbers that are not divisible by 2, 3, or 5, we subtract the count of numbers that are divisible by at least one of them from the total numbers (1 to 100):

\[ \text{Total numbers} = 100 \]
\[ \text{Not divisible by 2, 3, or 5} = 100 - N(2 \cup 3 \cup 5) = 100 - 74 = 26 \]

### Final Answer
The number of positive integers up to 100 that are not divisible by 2, 3, or 5 is **26**.

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