If r and s are positive integers, is the sum r + s an odd integer? (1) r \cdot s is an odd… — GMAT

If $r$ and $s$ are positive integers, is the sum $r + s$ an odd integer?

(1) $r \cdot s$ is an odd integer.
(2) $r$ is an even integer.

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) gives enough information to determine that

r + s

must be even, resulting in a definitive 'No' answer to the question. In Data Sufficiency, a definitive 'No' indicates sufficiency. Statement (2) does not fix the parity of

s

, so

r + s

could be either even or odd, making it insufficient.

Step-by-Step Solution

Analyze the question stem

We are given that

r

and

s

are positive integers and need to answer the Yes/No question: Is

r + s

odd?

Establishing the target question and constraints is essential before evaluating the statements.

Evaluate Statement (1):

r \cdot s

is an odd integer

The product of two integers is odd if and only if both integers are odd. Therefore,

r

is odd and

s

is odd. The sum of two odd integers is always an even integer (odd + odd = even). Thus,

r + s

is definitely not odd.

A definitive 'No' answer to a Yes/No Data Sufficiency question means the statement is SUFFICIENT.

Evaluate Statement (2):

r

is an even integer

r = 2

and

s = 1

, then

r + s = 3

(odd, answer is Yes). If

r = 2

and

s = 2

, then

r + s = 4

(even, answer is No). Because both Yes and No are possible, Statement (2) is INSUFFICIENT.

Since Statement (2) allows multiple outcomes for the target question, it does not provide sufficient information.

Key Concept

Parity rules for addition and multiplication of integers in Data Sufficiency Yes/No decision logic.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic