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Divisibility & Primes - GMAT QR question

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Divisibility & Primes - GMAT QR question

by adthedaddy » Tue Aug 07, 2012 8:44 pm
Following is the question from GMAT Quant Review 1st Edition (Pg.182, Solution to Ques. No. 86)

Q. 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 already a solution given in the QR but I am not able to understand the same.

Request my friends to plz help me with a simpler and understandable solution.
Thanks in advance :)
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Source: — Data Sufficiency |
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by niketdoshi123 » Tue Aug 07, 2012 10:12 pm
adthedaddy wrote:Following is the question from GMAT Quant Review 1st Edition (Pg.182, Solution to Ques. No. 86)

Q. 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 already a solution given in the QR but I am not able to understand the same.

Request my friends to plz help me with a simpler and understandable solution.
Thanks in advance :)
statement 1:

n>3
let n = 4
4 can be written as the sum of two positive numbers in two ways
4 = 2 + 2
4 = 1 + 3
In each of the above ways there is just 1 different prime number.
Hence n = 4 CANNOT be written as the sum of 2 different positive prime numbers. Ans is NO

Let n = 5
5 can be written as the sum of two positive numbers in two ways
5 = 2 + 3
5 = 4 + 1
We can see that n = 5 CAN be written as the sum of 2 different positive prime numbers. Ans is YES

Hence the statement is insufficient to answer the question.

Statement 2:

n is odd
We need to find whether n is sum of 2 different positive prime numbers

Here we have to apply two properties
1) odd + even = odd
odd + odd = even
even + even = even
2) there is only one even prime number, 2.

Hence , n (odd) = 2 + odd number
Let n = 11
11 = 2 + odd number => odd number = 9, but 9 is not a prime number.
Hence n = 11 cannot be written as the sum of 2 different prime numbers. Ans is NO

we proved in the above statement that n = 5 can be written as the sum of 2 different prime numbers. Ans is YES

hence the statement is insufficient to answer the question.

Combining both the statements

we can use both the examples used in statement 2 and prove insufficiency.

the correct answer is E
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