For how many integer values of k in the range 1 \le k \le 60 is the expression 3k^2 + 5k +… — GMAT

For how many integer values of $k$ in the range $1 \le k \le 60$ is the expression $3k^2 + 5k + 7$ an even integer?

Answer: 0

The expression

3k^2 + 5k + 7

can be evaluated for parity by considering the cases for

k

.
If

k

is even,

3k^2

is even,

5k

is even, and

7

is odd. The sum of two even integers and an odd integer (even + even + odd) is always odd.
If

k

is odd,

3k^2

is odd,

5k

is odd, and

7

is odd. The sum of three odd integers (odd + odd + odd) is always odd.
Because the expression yields an odd integer for every integer

k

, there are no integer values of

k

in the specified range for which the expression is even. Therefore, the correct count is 0.

Examine parity by testing even and odd cases for k

If k is even: 3(even)^2 + 5(even) + 7 = even + even + odd = odd. If k is odd: 3(odd)^2 + 5(odd) + 7 = odd + odd + odd = odd.

Covering both cases establishes the parity of the expression for all integer inputs.

Count the number of values of k in 1 <= k <= 60 that yield an even result

Since the expression is odd for all integer values of k, zero values yield an even integer.

The question specifically asks for the number of integer values of k that make the expression even.

Parity Rules for Addition, Multiplication, and Algebraic Expressions

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