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.

Cevap: Cevap

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

Soru