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

Options

A. 24
B. 25
C. 26
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 2, 3, and 5

1. **Count numbers divisible by 2:**
- The numbers divisible by 2 up to 100 are: 2, 4, 6, ..., 100.
- This forms an arithmetic series where the first term \(a = 2\) and the last term \(l = 100\) with a common difference \(d = 2\).
- The number of terms \(n\) can be calculated using the formula for the \(n\)-th term of an arithmetic series:
\[ n = \frac{l - a}{d} + 1 = \frac{100 - 2}{2} + 1 = 50 \]

2. **Count numbers divisible by 3:**
- The numbers divisible by 3 up to 100 are: 3, 6, 9, ..., 99.
- Here, \(a = 3\), \(l = 99\), and \(d = 3\).
- The number of terms:
\[ n = \frac{l - a}{d} + 1 = \frac{99 - 3}{3} + 1 = 33 \]

3. **Count numbers divisible by 5:**
- The numbers divisible by 5 up to 100 are: 5, 10, 15, ..., 100.
- Here, \(a = 5\), \(l = 100\), and \(d = 5\).
- The number of terms:
\[ n = \frac{l - a}{d} + 1 = \frac{100 - 5}{5} + 1 = 20 \]

### Step 2: Count the overlaps (numbers divisible by combinations of 2, 3, and 5)

4. **Count numbers divisible by 6 (2 and 3):**
- The numbers divisible by 6 up to 100 are: 6, 12, 18, ..., 96.
- Here, \(a = 6\), \(l = 96\), and \(d = 6\).
- The number of terms:
\[ n = \frac{l - a}{d} + 1 = \frac{96 - 6}{6} + 1 = 16 \]

5. **Count numbers divisible by 10 (2 and 5):**
- The numbers divisible by 10 up to 100 are: 10, 20, 30, ..., 100.
- Here, \(a = 10\), \(l = 100\), and \(d = 10\).
- The number of terms:
\[ n = \frac{l - a}{d} + 1 = \frac{100 - 10}{10} + 1 = 10 \]

6. **Count numbers divisible by 15 (3 and 5):**
- The numbers divisible by 15 up to 100 are: 15, 30, 45, ..., 90.
- Here, \(a = 15\), \(l = 90\), and \(d = 15\).
- The number of terms:
\[ n = \frac{l - a}{d} + 1 = \frac{90 - 15}{15} + 1 = 6 \]

7. **Count numbers divisible by 30 (2, 3, and 5):**
- The numbers divisible by 30 up to 100 are: 30, 60, 90.
- Here, \(a = 30\), \(l = 90\), and \(d = 30\).
- The number of terms:
\[ n = \frac{l - a}{d} + 1 = \frac{90 - 30}{30} + 1 = 3 \]

### Step 3: Apply the Inclusion-Exclusion Principle

Now we can apply the Inclusion-Exclusion principle to find the total count of numbers divisible by 2, 3, or 5:
\[ N(2 \cup 3 \cup 5) = N(2) + N(3) + N(5) - N(2 \cap 3) - N(2 \cap 5) - N(3 \cap 5) + N(2 \cap 3 \cap 5) \]
Substituting the values we found:
\[ N(2 \cup 3 \cup 5) = 50 + 33 + 20 - 16 - 10 - 6 + 3 = 74 \]

### Step 4: Calculate the numbers not divisible by 2, 3, or 5

Finally, to find the number of integers up to 100 that are not divisible by 2, 3, or 5, we subtract the count of those that are divisible from the total count of integers up to 100:
\[ \text{Total integers} = 100 \]
\[ \text{Not divisible by 2, 3, or 5} = 100 - N(2 \cup 3 \cup 5) = 100 - 74 = 26 \]

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

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