1998 CEMC Pascal Problems/Problem 6

Problem

If $x = 3$ , which of the following expressions is an even number?

$\text{ (A) }\ 9x \qquad\text{ (B) }\ x^3 \qquad\text{ (C) }\ 2(x^2 + 9) \qquad\text{ (D) }\ 2x^2 + 9 \qquad\text{ (E) }\ 3x^2$

Solution

We can see that answer choice C is always multiplied by $2$ , which makes the number even when $x$ is an integer. When $x = 3$ , choice C is $2(3^2 + 9) = 2(9 + 9) = 2 \times 18 = 36$ .

We can also verify that the other answer choices don't work:

$9 \times 3$ is odd because an odd multiplied by an odd number is always an odd number. We also get that $9 \times 3 = 27$ .

$3^3$ is odd, as an odd number to any power is always an odd number. We can also verify this with the value being $27$ .

$2 \times 3^2 + 9$ is odd, as the coefficient on the first number is even, so the first number is even. $9$ is odd, and an even and odd added together always results in an odd number. Evaluating this, we get $2 \times 9 + 9 = 18 + 9 = 27$ .

$3 \times 3^2$ is odd, since the coefficient isn't even, and an odd number raised to any power is an odd number. We get $3 \times 3^2 = 3^3 = 27$ too.

Thus, the answer is $\boxed {\textbf {(C) } 2(x^2 + 9)}$ .

~anabel.disher

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1998 CEMC Pascal Problems/Problem 6

Problem

Solution