If p, q, and r are integers such that 3p + 2q + r^2 = 17, and p is an odd integer, which o… — GMAT

If $p$ , $q$ , and $r$ are integers such that $3p + 2q + r^2 = 17$ , and $p$ is an odd integer, which of the following statements MUST be true?

r

is an even integer

Because

p

is odd,

3p

is odd. The term

2q

is always even for any integer

q

. Adding an odd and an even number gives an odd sum (

3p + 2q

). Since the total sum

3p + 2q + r^2

equals

17

, which is an odd number,

r^2

must be even. An integer whose square is even must be even itself, so the statement that

r

is an even integer must be true.

Analyze the parity of the term

3p

Since

p

is given as an odd integer,

\text{odd} \times \text{odd} = \text{odd}

, so

3p

is an odd integer.

Multiplying an odd integer by an odd integer yields an odd integer.

Analyze the parity of the term

2q

2q

is an even integer for any integer

q

Any integer multiplied by

2

produces an even integer.

Determine the parity of the sum

3p + 2q

3p + 2q = \text{odd} + \text{even} = \text{odd}

The sum of an odd integer and an even integer is always odd.

Evaluate the equation

(3p + 2q) + r^2 = 17

to find the parity of

r^2

and

r

\text{odd} + r^2 = 17\ (\text{odd}) \implies r^2

must be even

\implies r

must be an even integer.

For the sum of an odd integer and

r^2

to equal an odd integer (

17

r^2

must be even. An integer whose square is even must itself be even.

Parity rules of addition and multiplication of integers:

\text{odd} + \text{even} = \text{odd}

and

\text{even} \times \text{any integer} = \text{even}

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