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Re: Can the positive integer n be written as the sum of two different posi [#permalink]

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23 Dec 2015, 21:00

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the only way for n to be odd is even+odd = odd. the only even prime = 2. so then ten n = even prime + odd prime.??? whys that wrong?

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 some particular, fixed number. If we take two statements together the question becomes: can odd integer n, which is greater than 3, be written as the sum of two different prime numbers?

Now, if EVERY odd integer greater than 3 can be written as the sum of two different prime numbers, then taken together statements would be sufficient as we get definite YES answer to the question (because if it can be done for EVERY odd integer greater than 3 then it can be done for some particular n, from this group, too). Also, if NONE of the odd integers greater than 3 can be written as the sum of two different prime numbers, then taken together statements would still be sufficient, though at this time we'd getdefinite NO answer to the question (because if it cannot be done for ANY odd integer greater than 3 then it can not be done for some particular n, from this group, too).

Next, if we can find two values of odd integer n greater than 3 and one of them can be written as the sum of two different prime numbers and another cannot, then taken together statements would NOT be sufficient.

For this question the answer is E:

If n=5=odd>3, then the answer would be YES, 5=2+3=prime+prime;

If n=11=odd>3, then the answer would be NO, (11=odd and in order it to be the sum of two different primes one must be 2=even=prime, in this case another number would be 9, since 9 is not a prime, you cannot write 11 as the sum of two different primes).

So, we have two values of odd integer n greater than 3: one of them can be written as the sum of two different prime numbers and another cannot, hence taken together statements are not sufficient.

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