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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