CEMC Cayley

Algebra Difficulty 2.2 Junior Find the answer

An integer $x$ is chosen so that $3 x+1$ is an even integer. Which of the following must be an odd integer?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

If $x$ is an integer for which $3 x+1$ is even, then $3 x$ is odd, since it is 1 less than an even integer. If $3 x$ is odd, then $x$ must be odd (since if $x$ is even, then $3 x$ would be even). If $x$ is odd, then $7 x$ is odd (odd times odd equals odd) and so $7 x+4$ is odd (odd plus even equals odd). Therefore, the one expression which must be odd is $7 x+4$ .

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