If n is a positive integer, what is the greatest common factor

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Hi Bunuel what if the n is in the form of 2^p where none of the two factors of 2^p will add to a prime number. but we do not know what is the value of P so we certainly can not determine GCF .So statement 1 is not sufficient.

2^p is 2, 4, 8, ... For any of these values you can pick 1 and 2 as factors, which gives the sum of 3, which is a prime.

Dear Bunuel,

Can you elaborate more about the line

(1) No two different factors of n sum to a prime number. This implies that 2 is NOT a factor of n, if it were then the sum of two factors of n, 1 and 2, would be a prime number.

Any even number has at least the following factors: 1 and 2. The sum = 3 = prime.

(1) says that "No two different factors of n sum to a prime number". So, n is NOT even.

OFFICIAL EXPLANATION:

(1) SUFFICIENT: Every number has 1 as a factor. If n were an even integer, then 1 and 2 would both be factors of n. The sum of 1 and 2 is 3, though, which is prime. Therefore, because 1 has to be a factor, 2 cannot also be a factor. Therefore, n is odd, as are all factors of n (since an odd number can’t have an even factor).
The prime factorization of 64 is 2 , so 64 has no odd factors other than 1.
All factors of n are odd, and all factors of 64 are even except 1. The greatest common factor of n and 64 is therefore 1. The statement is sufficient.

(2) SUFFICIENT: 2,310 is an even integer. If n were an even integer, then the greatest common factor of n and 2,310 would be even (since n and 2,310 would have at least the factor 2 in common). Since the greatest common factor, 165, is odd, it follows that n cannot be even.
If you’re not sure about that, prove it to yourself. Break 2,310 and 165 into their prime factors:
2,310: 2, 3, 5, 7, 11
165: 3, 5, 11
Since 165 is the greatest common factor, n can’t contain a 2 or a 7. Therefore, n is not even.
Thus all factors of n are odd, and, as mentioned above, all factors of 64 are even except 1. The greatest common factor of n and 64 is therefore 1. The statement is sufficient.
The correct answer is (D).

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