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GMAT Practice Question (One of Hundreds)

X and Y are both positive integers whose combined factors include 3 and 7. Is the sum X + Y + 1 an odd integer?

Both X and Y are divisible by 2
X + 2 = Y

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: D

For X + Y + 1 to be odd, the sum X + Y must be even.

Do not be distracted by "X and Y are both positive integers whose combined factors include 3 and 7." The factors given do not allow you to conclude that X or Y is either odd or even. To conclude that X and Y are even, X and Y need to have at least one even factor. To conclude that X and Y are odd, X and Y must only have odd factors.
For X + Y + 1 to be odd, the sum X + Y must be even since adding one to an even integer makes it odd. Said algebraically:
X + Y + 1 = odd
X + Y = even
The sum of two integers will be even if and only if the parity of the two numbers is the same. In other words, odd + odd = even and even + even = even. However, the sum of two numbers of different parity is odd (i.e., odd + even = odd). Consequently, in order for X + Y = even, both X and Y must be of the same parity. There are two possibilities:
X_odd + Y_odd = even
X_even + Y_even = even
Evaluate Statement (1) alone.
1. A number is divisible by 2 if and only if it is even. Consider the following examples:
  4 is even and divisible by 2
  5 is not even and not divisible by 2
  6 is even and divisible by 2
  7 is not even and not divisible by 2
  8 is even and divisible by 2
  9 is not even and not divisible by 2
  10 is even and divisible by 2
  11 is not even and not divisible by 2
2. Since Statement (1) tells us that both X and Y are divisible by 2, both X and Y are even. Since X and Y have the same parity, the sum X + Y is even and the sum X + Y + 1 is odd; Statement (1) is SUFFICIENT.
3. Statement (1) alone is SUFFICIENT.
Evaluate Statement (2) alone.
1. If you take a number and add 2, you do not change the parity of that number. Consider the following examples:
  4 {i.e., even} + 2 = 6 {i.e., even}
  5 {i.e., odd} + 2 = 7 {i.e., odd}
  6 {i.e., even} + 2 = 8 {i.e., even}
  7 {i.e., odd} + 2 = 9 {i.e., odd}
  8 {i.e., even} + 2 = 10 {i.e., even}
  9 {i.e., odd} + 2 = 11 {i.e., odd}
2. Statement (2) indicates that the parity of X and Y are the same since adding 2 to X will not change the parity of X.
  X + 2 = Y
  Parity_X + 2 = Parity_Y
  Parity_X = Parity_Y
  Statement (2) is SUFFICIENT.
3. Statement (2) alone is SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.

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