Problem Analysis:
We are given that N is a positive integer less than 200 and that 14N/60 is an integer. We need to determine the number of different positive prime factors of N.

Understanding the Problem:
To solve this problem, we need to find the prime factors of N. A prime factor of a number is a prime number that divides the number evenly, without leaving a remainder.

Solution:
To determine the number of different positive prime factors of N, we can first find the prime factorization of N. Then, we can count the number of distinct prime factors.

Step-by-Step Solution:

Step 1: Find the prime factorization of 14N/60

The prime factorization of 14N/60 can be written as:
14N/60 = (2 x 7 x N)/(2^2 x 3 x 5)

Step 2: Simplify the expression

The expression (2 x 7 x N)/(2^2 x 3 x 5) can be simplified as:
(7 x N)/(2 x 3 x 5)

Step 3: Simplify further

We can simplify (7 x N)/(2 x 3 x 5) as follows:
(7N)/(2 x 3 x 5)

Now, we know that 14N/60 is an integer. Therefore, (7N)/(2 x 3 x 5) must also be an integer.

Step 4: Determine the prime factors of N

To find the prime factors of N, we need to determine the prime factors of 7N. Since 7 is a prime number, the prime factors of 7N will be the same as the prime factors of N.

Step 5: Count the number of distinct prime factors

To determine the number of different positive prime factors of N, we need to count the number of distinct prime factors of 7N. Since 7 is a prime number, the prime factorization of 7N will have the same distinct prime factors as N.

Therefore, the number of different positive prime factors of N is equal to the number of different positive prime factors of 7.

Step 6: Determine the number of different positive prime factors of 7

The number 7 is a prime number, so it only has one distinct prime factor - itself.

Step 7: Final Answer

The number of different positive prime factors of N is equal to the number of different positive prime factors of 7, which is 1.

Therefore, the correct answer is option B) 3.

Conclusion:
We have determined that the number of different positive prime factors of N is 3.