For all integers k and m, if k^2 + 3km + 2m^2 is an odd integer, then k + m must be an odd… — GMAT

For all integers $k$ and $m$ , if $k^2 + 3km + 2m^2$ is an odd integer, then $k + m$ must be an odd integer.

Answer: Answer

The statement is True.

The statement is true because factoring

k^2 + 3km + 2m^2

yields

(k + m)(k + 2m)

. By the fundamental properties of parity, a product of two integers is odd if and only if both factors are odd. Consequently,

k + m

must be odd.

Factor the algebraic expression

k^2 + 3km + 2m^2

k^2 + 3km + 2m^2 = (k + m)(k + 2m)

Factoring into a product of linear terms allows us to apply parity rules for multiplication.

Apply parity rules of multiplication to the product

(k + m)(k + 2m)

Since

(k + m)(k + 2m)

is odd, both

(k + m)

and

(k + 2m)

must be odd integers.

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

Conclude the parity of

k + m

k + m

is guaranteed to be an odd integer.

It is one of the two factors required to be odd for the product to be odd.

Parity Rules for Integer Products and Factoring

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