Amrabdelnaby wrote:If x and y are integers, what is the remainder when x² + y² is divided by 5?
(1) When x - y is divided by 5, the remainder is 1.
(2) When x + y is divided by 5, the remainder is 2.
First, neither one of the statements looks sufficient. Maybe plug in some numbers to confirm.
Statement 1: When x - y is divided by 5, the remainder is 1.
Try 7 and 1.
x² + y² = 49 + 1 = 50 Remainder when divided by 5 is 0.
Try 8 and 2.
x² + y² = 64 + 4 = 68 Remainder when divided by 5 is 3.
Insufficient.
Statement 2: When x + y is divided by 5, the remainder is 2.
Try 5 and 2.
x² + y² = 25 + 4 = 29 When divided by 5, remainder is 4.
Try 4 and 3.
x² + y² = 16 + 9 = 25 When divided by 5, remainder is 0.
Insufficient.
Now notice the following.
From Statement 1: (x - y)² = x² - 2xy + y²
From Statement 2: (x + y)² = x² + 2xy + y²
So if we add, we get (x - y)² + (x + y)² = 2x² + 2y²
Almost there.
Convert the statements into math.
(1) x - y = 5k + 1
(2) x + y = 5l + 2
Square both statements and add.
x² - 2xy + y² = 25k² + 10k + 1
x² + 2xy + y² = 25l² + 10l + 4
2x² + 2y² = 25k² + 10k + 10l + 5
All the terms on the right are divisible by 5, and x and y are integers.
So if 2x² + 2y² is divisible by 5, then x² + y² is divisible by 5.
The correct answer is
C.
Alternate Method
Combine the statements and find some numbers that work for both. Then plug into x² + y².
(1) x - y = 5k + 1 --> x = 9 and y = 3 works.
(2) x + y = 5l + 2 --> x = 9 and y = 3 works.
x² + y² = 81 + 9 = 100 When divided by 5, remainder is 0.
(1) x - y = 5k + 1 --> x = 4 and y = 3 works.
(2) x + y = 5l + 2 --> x = 4 and y = 3 works.
x² + y² = 16 + 9 = 25 When divided by 5, remainder is 0.
(1) x - y = 5k + 1 --> x = 14 and y = 3 works.
(2) x + y = 5l + 2 --> x = 14 and y = 3 works.
x² + y² = --6 + 9 = --5 When divided by 5, remainder is 0.
Only numbers such that x ends in 9 or 4 and y ends in 3 seem to fit both statements, and x² + y² is always divisible by 5.
So it seems safe to say that combined the statements are sufficient.
The correct answer is
C.
