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
smallest n = 2
n = 2 ---> 2,4,6 nope
n = 3 ---> 3,5,7 YES
n = 5 ---> 5,7,9 nope
n = 7 ---> 7,9,11 nope
n = 11 ---> 11,13,15 nope
..
n = 23 --> 23, 25, 27, nope
Other than 2, all primes are odd
so we need 3 sequential odd numbers to have the above property
BUT, since one of every three sequential odd numbers is a multiple of three, at least one of them can't be prime
so 3,5,7 is the only one
there is 1 of them
To determine how many different positive integers n will result in n, n+2, and n+4 representing three different prime numbers, we need to check the conditions for each option.
If n is an even number greater than 2, then n+2 will be an even number greater than 4, and therefore not prime. Therefore, n cannot be even.
If n is an odd number greater than 1, then it cannot be divisible by 3 because n, n+2, and n+4 would all have a remainder of 0 when divided by 3, making them not prime. Therefore, n cannot be divisible by 3.
Combining these conditions, n must be an odd number greater than 1 and not divisible by 3.
The possible values for n are:
n = 5 ⇒ n+2 = 7, n+4 = 9 (not prime)
n = 7 ⇒ n+2 = 9 (not prime)
n = 11 ⇒ n+2 = 13, n+4 = 15 (not prime)
n = 13 ⇒ n+2 = 15 (not prime)
n = 17 ⇒ n+2 = 19, n+4 = 21 (not prime)
n = 19 ⇒ n+2 = 21 (not prime)
Therefore, there are no values of n that satisfy the conditions, so the answer is A. none.
To find the number of different positive integers n that satisfy the given conditions, we need to check each value of n.
We know that a prime number is only divisible by 1 and itself. Therefore, we need to check whether n, n+2, and n+4 are prime numbers.
Checking for n = 1:
n = 1, n+2 = 3, n+4 = 5
All three numbers n, n+2, and n+4 are prime numbers.
Checking for n = 2:
n = 2, n+2 = 4, n+4 = 6
Only the number n is a prime number. n+2 and n+4 are not prime.
Checking for n = 3:
n = 3, n+2 = 5, n+4 = 7
All three numbers n, n+2, and n+4 are prime numbers.
Checking for n = 4:
n = 4, n+2 = 6, n+4 = 8
None of the numbers n, n+2, and n+4 are prime.
Checking for n = 5:
n = 5, n+2 = 7, n+4 = 9
Only the number n is a prime number. n+2 and n+4 are not prime.
We have exhausted all the possible values for n up to this point, and we have found that there are two different positive integers, n = 1 and n = 3, where n, n+2, and n+4 represent three different prime numbers.
Therefore, the answer is C. 2