If x and y are positive integers, is x + y divisible by 3? (1) x^2 + y^2 is divisible by 3… — GMAT

If $x$ and $y$ are positive integers, is $x + y$ divisible by 3?

(1) $x^2 + y^2$ is divisible by 3.
(2) $x^2 - y^2$ is divisible by 3.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Answer
B
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (1) alone is sufficient because the square of any integer leaves a remainder of 0 or 1 when divided by 3. The only way for

x^2 + y^2

to be a multiple of 3 is if both

x^2

and

y^2

leave remainders of 0, meaning both

x

and

y

are multiples of 3. Therefore,

x + y

must be a multiple of 3. Statement (2) alone is not sufficient because

x^2 - y^2 = (x - y)(x + y)

being divisible by 3 allows

x + y

to be divisible by 3 (e.g.,

x = 2, y = 1

) or not divisible by 3 (e.g.,

x = 4, y = 1

Step-by-Step Solution

Analyze remainder properties of perfect squares modulo 3.

For any integer

z

z^2 \equiv 0 \pmod 3

z

is a multiple of 3, and

z^2 \equiv 1 \pmod 3

z

is not a multiple of 3.

Establishing the possible values of perfect squares modulo 3 simplifies the evaluation of both statements.

Evaluate Statement (1):

x^2 + y^2

is divisible by 3.

Since

x^2 \pmod 3 \in \{0, 1\}

and

y^2 \pmod 3 \in \{0, 1\}

, the sum

x^2 + y^2 \equiv 0 \pmod 3

can only occur if

x^2 \equiv 0 \pmod 3

and

y^2 \equiv 0 \pmod 3

. This implies both

x

and

y

are divisible by 3, so

x + y

must also be divisible by 3.

Statement (1) yields a definitive 'Yes' answer, making it sufficient.

Evaluate Statement (2):

x^2 - y^2

is divisible by 3.

Factor as

(x - y)(x + y)

. If

x = 2

and

y = 1

x^2 - y^2 = 3

(divisible by 3) and

x + y = 3

(divisible by 3, answer Yes). If

x = 4

and

y = 1

x^2 - y^2 = 15

(divisible by 3) and

x + y = 5

(not divisible by 3, answer No).

Since statement (2) yields both 'Yes' and 'No' outcomes, it is not sufficient.

Key Concept

Quadratic Remainder Constraints in Data Sufficiency

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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