If x, y, and z are integers such that x^2 + y^2 + z^2 is an odd integer, then the sum x + … — GMAT

If $x$ , $y$ , and $z$ are integers such that $x^2 + y^2 + z^2$ is an odd integer, then the sum $x + y + z$ must be an odd integer.

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The statement is True because the square of any integer preserves the parity of that integer, making the parity of

x^2 + y^2 + z^2

identical to the parity of

x + y + z

The statement is true because for all integers

k

, the parity of

k^2

is identical to the parity of

k

. Consequently, the parity of the sum of squares

x^2 + y^2 + z^2

is always identical to the parity of the sum

x + y + z

. If the sum of squares is odd, the sum of the variables must also be odd.

Analyze the parity relationship between any integer

k

and its square

k^2

k

is even,

k^2

is even; if

k

is odd,

k^2

is odd. (Note that zero is an even integer, and

0^2 = 0

is even).

Multiplying an even integer by itself yields an even number, and multiplying an odd integer by itself yields an odd number.

Relate the parity of

x^2 + y^2 + z^2

x + y + z

The expression

x^2 + y^2 + z^2

has the exact same parity as

x + y + z

Replacing each squared term with its base term does not change whether the sum is even or odd.

Apply the given condition to deduce the final parity.

Since

x^2 + y^2 + z^2

is odd,

x + y + z

must be odd.

Two expressions with identical parity must both be odd if one is given as odd.

Parity Invariance of Integer Powers

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