For how many different positive integers n will n, n+2, and n+4 represent three different prime numbers?
A. none
B. 1
C. 2
D. 3
To solve this problem, we need to check whether n, n+2, and n+4 represent three different prime numbers for different positive integers n.
To determine if a number is prime, we need to check if it is only divisible by 1 and itself. We can do this by checking if any number less than or equal to the square root of the number evenly divides it.
Let's examine each possible positive integer value for n:
1. If n = 1, then n+2 = 3 and n+4 = 5. These are three different prime numbers. Hence, when n = 1, the answer is 1.
2. If n = 2, then n+2 = 4 and n+4 = 6. None of these numbers are prime as both 4 and 6 are divisible by 2. Therefore, when n = 2, the answer is none.
3. If n = 3, then n+2 = 5 and n+4 = 7. These are three different prime numbers. Hence, when n = 3, the answer is 1.
For n > 3, as we increase n, the subsequent numbers n+2 and n+4 will always be even. For even numbers greater than 2, it is not possible for them to be prime, as they are divisible by 2. Therefore, for n > 3, there are no more possible values for which n, n+2, and n+4 represent three different prime numbers.
Therefore, the answer is 1 (option B) since when n = 1 and n = 3, n, n+2, and n+4 represent three different prime numbers.