Can the positive integer n be written as the sum of two different posi

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.
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From 1: n>3 => put n=4 (Cannot be written), n=5 (can be written), Insufficient, A,D ruled out
From 2: n= odd => put n=1(Cannot be written), n=5 (can be written), Insufficient, B ruled out
Combining 1 &2 :
n>3 and n is odd => sum of the primes is odd => one of the primes =2
Now, to rephrase this: the question asks "odd no. - 2 = prime ?" => maybe and may not be : C ruled out

Answer is E

I really don't like the word 'Can' in this question as it is not precise. Is there a chance that this would be a real GMAT question?
I mean, I understand your explanation Bunuel, but you can actually answer the question with either statements. "Can it be?" - Sure it can, but also cannot, it depends on what the value of 'n' is.

Is that the way to approach this question? Ask yourself: "What is the value of n?" and if this is not given in the statements, then choose E?

This is OG question, so it's quite "real".
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2 different prime numbers : We are ruling out option 'C' with an eg 11= 2+9 (9 is not prime)

However 11 = 2 +3 +3 = Sum of 2 different prime nos (2 and 3).based on this can we select C ...???

The question asks whether n can be written as the sum of two different prime numbers, so whether n = prime 1 + prime 2. If n = 11, then it cannot be written as the sum of two different primes. 2 + 3 + 3 + 3 cannot be said to be the sum of two primes, it's the sum of 4 numbers not 2.
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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

n=5 gives an YES answer to the question. 5 CAN be written as the sum of two difference prime numbers: 5 = 2 + 3.

While if n=11 or n=17 the answer would be NO. Neither 11 nor 17 can be written as the sum of two different prime numbers.
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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a 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.

There is one variable (n) and 2 equations are given from the 2 conditions, so there is high chance (D) will be our answer.
For condition 1, the answer is 'yes' for n=5=2+3, but 'no' for 23=2+21
For condition 2, the answer is 'yes' for n=5=2+3, but 'no' for 23=2+21
Looking at the conditions together, answer is 'yes' for n=5=2+3, but 'no' for 23=2+21.
The answer is not unique; the answer becomes (E).

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.

Hi VeritasKarishma - just to make sure my understanding is correct

-- n = 6 or n = 17 would be examples of "NO" as it is not possible for n =6 or n = 17 to be the sum of two prime numbers

-- n = 7, 8 / 9 / 10 would ALL be examples of "YES", it is possible to these values of n to be the sum of two prime numbers.

- i am thinking off a weird number like n = 101 would a "No" as well because if you need two prime numbers, one of these two has to be "2" (given odd prime + odd prime will give an even number always) and 2 + 99 is the only combination to reach 101

Given 99 is not prime, n = 101 would be a "No"

I believe that is always goin

n cannot be even.
We know that n is odd as per stmnt 2.
So n = 5 is a possible value which can be written as 2 + 3 so answer here will be "yes"
n = 11 is a possible value which can be written as 2 + 9 so answer here will be "no".

The point to note here is that if n is odd and written as sum of two primes, one prime must be odd and the other even. The only even prime is 2.

So to check for any odd value of n, you need to look whether (n - 2) is prime or not.
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Hi everyone,

Many thanks to all of you for the explanations.

Since the question uses the word "Can" and not "Must" or "Is" for example, aren't we only supposed to see if there is at least ONE number that can satisfy the question ? (Instead of finding if it works for all numbers).

Thank you in advance,

Bunuel
VeritasKarishma

You are misinterpreting the role of CAN.

Can n be written as the sum of two different positive prime numbers?

To answer this, you need the value of n.

For example, if n = 10, you CAN write it as 3 + 7
But if n = 4, you CANNOT write it as sum of any two different primes.

What if we do not have the value of n? What information would be sufficient to know whether it CAN be written as sum of two different primes?

If we know that n is odd and greater than 3, is it enough information?

n can be 5. Then n CAN be written as 2 + 3
n can be 11. Then n CANNOT be written as sum of two different primes.
and so on..

So the information is not enough.
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wording of the questin is decieving, it is CAN and not IS.
like ofcourse n CAN be sum of different positive integers in both the cases

Here I am a little bit confused because " can be " is not strict requirement

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Think the easiest way is to pick numbers.
Lets pick 15 and 17.

(1) n is greater than 3.

15 satisfies the condition, 15 = 13+2. 17 does not satisfy

(2) n is odd.
again same explanation as above.


Combining both statements, again its not sufficient.