In a set of first 180 natural numbers find the number of integers, which are divisible by neither 2 nor 3 nor 5.

Options

A. 14
B. 15
C. 48
D. 51

Answer

To solve the problem of finding the number of integers in the first 180 natural numbers that are divisible by neither 2, 3, nor 5, we can use the principle of complementary counting. Here’s a step-by-step solution:

### Step 1: Count the total numbers
We start with the total number of natural numbers in the set, which is 180.

### Step 2: Count numbers divisible by 2
To find how many numbers are divisible by 2, we use the formula:
\[ \text{Count of numbers divisible by 2} = \left\lfloor \frac{180}{2} \right\rfloor = 90 \]

### Step 3: Count numbers divisible by 3
Next, we count how many numbers are divisible by 3:
\[ \text{Count of numbers divisible by 3} = \left\lfloor \frac{180}{3} \right\rfloor = 60 \]

### Step 4: Count numbers divisible by 5
Now, we count how many numbers are divisible by 5:
\[ \text{Count of numbers divisible by 5} = \left\lfloor \frac{180}{5} \right\rfloor = 36 \]

### Step 5: Count numbers divisible by combinations of 2, 3, and 5
We need to apply the principle of inclusion-exclusion to avoid double counting. We will count the numbers divisible by the least common multiples (LCMs) of the pairs:

- **Divisible by 6 (LCM of 2 and 3)**:
\[ \text{Count of numbers divisible by 6} = \left\lfloor \frac{180}{6} \right\rfloor = 30 \]

- **Divisible by 10 (LCM of 2 and 5)**:
\[ \text{Count of numbers divisible by 10} = \left\lfloor \frac{180}{10} \right\rfloor = 18 \]

- **Divisible by 15 (LCM of 3 and 5)**:
\[ \text{Count of numbers divisible by 15} = \left\lfloor \frac{180}{15} \right\rfloor = 12 \]

- **Divisible by 30 (LCM of 2, 3, and 5)**:
\[ \text{Count of numbers divisible by 30} = \left\lfloor \frac{180}{30} \right\rfloor = 6 \]

### Step 6: Apply the inclusion-exclusion principle
Now we can apply the inclusion-exclusion principle:
\[ \text{Total divisible by 2, 3, or 5} = (90 + 60 + 36) - (30 + 18 + 12) + 6 \]
Calculating this gives:
\[ = 186 - 60 + 6 = 132 \]

### Step 7: Count numbers not divisible by 2, 3, or 5
Finally, we subtract the count of numbers divisible by 2, 3, or 5 from the total count of natural numbers:
\[ \text{Count of numbers not divisible by 2, 3, or 5} = 180 - 132 = 48 \]

### Final Answer
Thus, the number of integers in the first 180 natural numbers that are divisible by neither 2, 3, nor 5 is **48**.