Practice GMAT Data Sufficiency Question

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x is a positive integer; what is the value of x?

The sum of any two positive factors of x is even
x is a prime number and x < 4

A)	Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B)	Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C)	BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D)	EACH statement ALONE is sufficient.
E)	Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Correct Answer: C

Evaluate Statement (1) alone.
1. Statement (1) says that the sum of any two factors is even. The sum of two integers is even under two circumstances:
  odd + odd = even
  even + even = even
2. Since the sum of any two factors is even, all the factors must have the same parity. If x had both even and odd factors, then it would be possible for two factors to add together and be odd (remember that an odd number + an even number = an odd number and, in Statement 1, the sum of any two positive factors must be even).
3. Since the problem says "the sum of any two positive factors of x is even" and the number 1 is a factor of any number, x must only contain odd factors. If x contained one even factor, it would be possible to add that even factor with the number one and the result would be an odd number. Since the number 2 is a factor of every even number, x cannot be even. Otherwise, it would be possible to add the factors 1 and 2 together and their sum would not be even.
4. Statement (1), when inspected carefully, says that x is an odd number that only contains odd factors. Since there are many possibilities (x = 1, 3, 5, 7, 9, 11, 15, ...), Statement (1) is NOT SUFFICIENT.
Evaluate Statement (2) alone.
1. Statement (2) says that x is a prime number less than 4. Remember that x must also be a positive integer (as per the original question). Although this narrows the possibilities for x, because there are still two possibilities (x = 2 or x = 3; both these values are prime, less than 4, and positive integers), Statement (2) is NOT SUFFICIENT. Please remember that the number one is not prime.
Evaluate Statements (1) and (2) together.
1. Statements (1) and (2), when taken together, definitively show that x = 3. Statements (1) and (2), when taken together, are SUFFICIENT. Answer choice C is correct.

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