Practice GMAT Data Sufficiency Question
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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
Correct Answer: D
 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.
 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  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.
 Statement (1) alone is SUFFICIENT.
 A number is divisible by 2 if and only if it is even. Consider the following examples:

Evaluate Statement (2) alone.
 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}  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.  Statement (2) alone is SUFFICIENT.
 If you take a number and add 2, you do not change the parity of that number. Consider the following examples:
 Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.
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