## Values of Integers

If $m$ and $n$ are integers such that $2m-n=3$, what are the possible values of $m-2n$?
(a) $-3$ only
(b) $0$ only
(c) only multiples of $3$
(d) any integer
(e) none of these
Source: NCTM Mathematics Teacher, September 2006

Solution
Let $x$ be an integer such that $x=m-2n$. We have the following two equations
$2m-n=3\qquad\qquad (1)$
$m-2n=x\qquad\qquad (2)$
Multiply Eq. $(1)$ by $-2$ and add to Eq. $(2)$
$-4m+2n=-6$
$m-2n=x$
————————
$-3m=x-6$
Divide both sides by $-3$
$m=-x/3+2$
For $m$ to be an integer, $x$ must be a multiple of $3$.

Answer: (c) only multiples of $3$

Alternative solution
$m-2n=m-2n+(3)-3$
$=m-2n+(2m-n)-3$
$=3m-3n-3$
$=3(m-n-1)$

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## About mvtrinh

Retired high school math teacher.
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