Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 2.5 Junior Find the answer

Suppose that $N = 3x + 4y + 5z$ , where $x$ equals 1 or -1, and $y$ equals 1 or -1, and $z$ equals 1 or -1. How many of the following statements are true? - $N$ can equal 0. - $N$ is always odd. - $N$ cannot equal 4. - $N$ is always even.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

When $N = 3x + 4y + 5z$ with each of $x, y$ and $z$ equal to either 1 or -1, there are 8 possible combinations of values for $x, y$ and $z$ . From this information, $N$ cannot equal 0, $N$ is never odd, $N$ can equal 4, and $N$ is always even. Therefore, exactly one of the four given statements is true.

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