In a set of first 350 natural numbers find the number of integers, which are not divisible by 5 or 7.

Options

A. 240
B. 120
C. 150
D. 300

Answer

To solve the problem of finding the number of integers in the first 350 natural numbers that are not divisible by 5 or 7, we can follow these steps:

### Step 1: Count the total natural numbers
The total number of natural numbers from 1 to 350 is simply 350.

### Step 2: Count the numbers divisible by 5
To find how many numbers are divisible by 5, we divide 350 by 5:
\[ \text{Numbers divisible by 5} = \frac{350}{5} = 70 \]

### Step 3: Count the numbers divisible by 7
Next, we count how many numbers are divisible by 7 by dividing 350 by 7:
\[ \text{Numbers divisible by 7} = \frac{350}{7} = 50 \]

### Step 4: Count the numbers divisible by both 5 and 7 (i.e., divisible by 35)
Since some numbers are counted twice (those that are divisible by both 5 and 7), we need to find how many numbers are divisible by 35 (the least common multiple of 5 and 7):
\[ \text{Numbers divisible by 35} = \frac{350}{35} = 10 \]

### Step 5: Apply the principle of inclusion-exclusion
Now we can use the principle of inclusion-exclusion to find the total number of integers that are divisible by either 5 or 7:
\[ \text{Numbers divisible by 5 or 7} = (\text{Numbers divisible by 5}) + (\text{Numbers divisible by 7}) - (\text{Numbers divisible by both 5 and 7}) \]
\[ = 70 + 50 - 10 = 110 \]

### Step 6: Calculate the numbers not divisible by 5 or 7
Finally, we subtract the count of numbers divisible by 5 or 7 from the total count of natural numbers:
\[ \text{Numbers not divisible by 5 or 7} = 350 - 110 = 240 \]

### Conclusion
Thus, the number of integers in the first 350 natural numbers that are not divisible by 5 or 7 is **240**.

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