The given problem asks us to find the number of positive integers less than 175 that are relatively prime to 3, 5, and 7. To solve this problem, we can use the principle of inclusion-exclusion.

Principle of Inclusion-Exclusion:
The principle of inclusion-exclusion states that if we want to count the number of elements in the union of several sets, we can start by summing the sizes of each individual set. However, we have to subtract the sizes of the intersections of all possible pairs of sets, then add back the sizes of the intersections of all possible triples of sets, and so on.

Step 1: Find the numbers relatively prime to 3:
To find the numbers relatively prime to 3, we need to count all the positive integers less than 175 that are not divisible by 3. We can use the formula for counting the positive integers less than a given number N that are not divisible by a prime number P:

Number of positive integers less than N that are relatively prime to P = N - N/P

Using this formula, the number of positive integers less than 175 that are relatively prime to 3 is:
175 - 175/3 = 175 - 58 = 117

Step 2: Find the numbers relatively prime to 5:
Similarly, the number of positive integers less than 175 that are relatively prime to 5 is:
175 - 175/5 = 175 - 35 = 140

Step 3: Find the numbers relatively prime to 7:
The number of positive integers less than 175 that are relatively prime to 7 is:
175 - 175/7 = 175 - 25 = 150

Step 4: Apply the principle of inclusion-exclusion:
Now, we need to apply the principle of inclusion-exclusion to find the numbers that are relatively prime to all three numbers (3, 5, and 7).

Number of positive integers less than 175 that are relatively prime to 3, 5, and 7 = Number relatively prime to 3 - Number relatively prime to 5 - Number relatively prime to 7 + Number relatively prime to both 5 and 7

117 - 140 - 150 + 175/35 = 117 - 140 - 150 + 5 = -178

However, since we are only interested in positive integers, the answer cannot be negative. Therefore, the correct answer is 175 - (-178) = 353.

Conclusion:
The number of positive integers less than 175 that are relatively prime to 3, 5, and 7 is 353.

The number of positive integers less than 175 that are divisible by 3 = 58 [175/3]