Consider two integers, x and y, where the dif… | GMAT Sprint

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PS · Problem Solving

sub-505

Number Properties

Consider two integers, x and y, where the difference x − y is odd. Which of the following statements is necessarily true?

I. The product xy is even.
II. The sum of their squares, $x^2 + y^2$ , is odd.
III. The square of their sum, $(x + y)^2$ , is even.

Answer Choices

Correct answer marked below

I only

II only

III only

I and II only

Correct

I, II, and III

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Explanation

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Official Explanation

Official

Given that x - y is odd, it follows that one of the integers must be even and the other must be odd.

I. The product xy is even. This is true since an even number multiplied by any integer results in an even product. For example, if x = 4 (even) and y = 3 (odd), then

xy = 4 * 3 = 12

(even).

II. The sum of their squares,

x^2 + y^2

, is odd. This is also true. For instance, if x = 4 and y = 3, then

x^2 + y^2 = 4^2 + 3^2 = 16 + 9 = 25

(odd).…

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Comments

radhikaApr 4, 2024

Love how you solved this

TanishaJul 28

wait, why is III wrong? (x+y)^2 being even seemed right to me at first lol

RosannaAug 4

if x-y is odd then x+y is also odd, so squaring an odd gives odd, not even

NayanJul 21

easy level for sure. just plug in like x=2, y=1 and check each roman numeral. only I and II survive

MariaApr 27

got confused between II and III for a sec. x^2+y^2 with one even one odd always comes out odd, but the sum squared is a different story

TanyaMay 11

same here, the (x+y)^2 vs x^2+y^2 distinction tripped me up

What this question tests

What it tests

Your command of divisibility, primes, factors, parity, and how the GMAT tests properties of integers.

Common trap

Forgetting that 1 is not prime, that 0 is divisible by every integer, or that negative numbers flip your assumptions.

Consider two integers, x and y, where the difference x − y is odd. Which of the following statements is necessarily true? I. The product xy is even. II. The sum of their squares, x2+y2x^2 + y^2x2+y2, is odd. III. The square of their sum, (x+y)2(x + y)^2(x+y)2, is even.