CEMC Fermat, 2020

Algebra Difficulty 2.5 Junior Find the answer Canada

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.

$0$
$1$
$2$
$3$
$4$

Your answer

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

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 $z$ :

$x$	$y$	$z$	$N$
1	1	1	12
1	1	-1	2
1	-1	1	4
1	-1	-1	-6
-1	1	1	6
-1	1	-1	-4
-1	-1	1	-2
-1	-1	-1	-12

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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