Bunuel wrote:
SOLUTION
Can the positive integer n be written as the sum of two different positive prime numbers?
(1) n is greater than 3.
(2) n is odd.
Note that n is a specific, fixed number. If we combine the two statements, the question becomes: can an odd integer n that is greater than 3, be expressed as the sum of two different prime numbers?
Now, if EVERY odd integer greater than 3 can be expressed as the sum of two different prime numbers, then the combined statements would be sufficient, giving a definite YES answer to the question (because if it is possible for EVERY odd integer greater than 3, it would also be possible for any particular n from this group). Similarly, if NONE of the odd integers greater than 3 can be expressed as the sum of two different prime numbers, the combined statements would still be sufficient, though in this case we would get a definite NO answer to the question (because if it is not possible for ANY odd integer greater than 3, then it would not be possible for any particular n from this group).
However, if we can find two values of odd integer n greater than 3 where one can be expressed as the sum of two different prime numbers and the other cannot, then the combined statements would NOT be sufficient.
For this question the answer is E:
If (n = 5 = odd) > 3, then the answer would be YES, as 5 = 2 + 3 = prime + prime;
If (n = 11 = odd) > 3, then the answer would be NO, (since 11 = odd and in order for it to be the sum of two different primes, one must be 2 = even = prime, in this case the other number would be 9, and since 9 is not a prime, 11 cannot be expressed as the sum of two different primes).
So, we have two values of the odd integer n greater than 3: one of them can be expressed as the sum of two different prime numbers and the other cannot, hence the combined statements are not sufficient.
Answer: E.
Hi Bunuel, I've 1 question regarding picking numbers for combined statement (1) + (2)
Could we also pick following numbers here: ??
n=5 -> 3+2 Yes
n=5 -> 4+1 No