eybrj2 wrote:
If x, y, and d are integers and d is odd, are both x and y divisible by d ?
(1) x + y is divisible by d.
(2) x − y is divisible by d.
Are x and y both divisible by d?
If d=1, the answer to the question stem is YES, since every integer is divisible by 1.
The only question is whether the answer to the question stem can be NO.
Statement 1:
Case 1: d=3, x=1 and y=2
In this case, x and y are NOT divisible by d, so the answer to the question stem is NO.
Since the answer can be YES (if d=1) or NO (as in Case 1), INSUFFICIENT.
Statement 2:
Case 2: d=3, x=4 and y=1
In this case, x and y are NOT divisible by d, so the answer to the question stem is NO.
Since the answer can be YES (if d=1) or NO (as in Case 2), INSUFFICIENT.
Statements combined:
Let d=3.
Implication:
x+y is a multiple of 3, with the result that x+y = 3, 6, 9, 12, 15, 18, 21...
x-y is a multiple of 3, with the result that x-y = 3, 6, 9, 12, 15, 18, 21...
Since the sum of x+y and x-y is equal to 2x, the sum of x+y and x-y must be EVEN.
Case 3: x+y=3 and x-y=3, implying that x=3 and y=0
In this case, both x and y are divisible by d=3.
Case 4: x+y=12 and x-y=6, implying that x=9 and y=3
In this case, both x and y are divisible by d=3.
Case 5: x+y=21 and x-y=9, implying that x=15 and y=6
In this case, both x and y are divisible by d=3.
In every case, x and y are both divisible by d.
Implication:
The answer to the question stem is YES.
SUFFICIENT.