GMAT Math

If x and y are integers, what is the remainder when x^2 + y^2 is divided by 5?

1) x-y divided by 5 the remainder is 1

2) x+y divided by 5 the remainder is 2

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.