If m and n are integers such that $(m+2n)(2m+n)=27$ , then the maximum possible value of is $2m-3n$

The correct answer is 17.

To find the maximum possible value of the given expression, let us first simplify our work by defining two new variables based on the factors in the given equation. Let:

$A=m+2n$

$B=2m+n$

Substituting these variables back into the original equation gives us:

$A\times B=27$

Since we are given that $m$ and $n$ are integers, the combinations $m+2n$ and $2m+n$ must also result in integers. Therefore, $A$ and $B$ must be an integer factor pair of 27. The possible integer pairs for $(A,B)$ that multiply to 27 are:

(1, 27), (27, 1), (-1, -27), (-27, -1), (3, 9), (9, 3), (-3, -9), and (-9, -3).

Next, we need to express $m$ and $n$ in terms of $A$ and $B$. We have the following system of linear equations:

$m+2n=A$

$2m+n=B$

We can solve for $n$ by eliminating $m$. If we multiply the first equation by 2, we get $2m+4n=2A$. Subtracting the second equation from this result gives:

$(2m+4n)-(2m+n)=2A-B$

$3n=2A-B$

$n=\frac{2A-B}{3}$

Following a similar process to solve for $m$ (multiplying the second equation by 2 and subtracting the first), we find:

$m=\frac{2B-A}{3}$

Because $m$ and $n$ must be integers, the numerators $2A-B$ and $2B-A$ must both be divisible by 3. Let us evaluate all our potential $(A,B)$ pairs to find valid integer values for $m$ and $n$:

Case 1: $(A,B)=(1,27)$
$n=\frac{2\left(1\right)-27}{3}=-\frac{25}{3}$ (Not an integer, so we reject this pair)

Case 2: $(A,B)=(27,1)$
$n=\frac{2\left(27\right)-1}{3}=\frac{53}{3}$ (Not an integer, reject)

Case 3: $(A,B)=(-1,-27)$
$n=\frac{2(-1)-(-27)}{3}=\frac{25}{3}$ (Not an integer, reject)

Case 4: $(A,B)=(-27,-1)$
$n=\frac{2(-27)-(-1)}{3}=-\frac{53}{3}$ (Not an integer, reject)

Case 5: $(A,B)=(3,9)$
$n=\frac{2\left(3\right)-9}{3}=\frac{-3}{3}=-1$
$m=\frac{2\left(9\right)-3}{3}=\frac{15}{3}=5$
This pair gives valid integers $(m,n)=(5,-1)$.

Case 6: $(A,B)=(9,3)$
$n=\frac{2\left(9\right)-3}{3}=\frac{15}{3}=5$
$m=\frac{2\left(3\right)-9}{3}=\frac{-3}{3}=-1$
This pair gives valid integers $(m,n)=(-1,5)$.

Case 7: $(A,B)=(-3,-9)$
$n=\frac{2(-3)-(-9)}{3}=\frac{3}{3}=1$
$m=\frac{2(-9)-(-3)}{3}=\frac{-15}{3}=-5$
This pair gives valid integers $(m,n)=(-5,1)$.

Case 8: $(A,B)=(-9,-3)$
$n=\frac{2(-9)-(-3)}{3}=\frac{-15}{3}=-5$
$m=\frac{2(-3)-(-9)}{3}=\frac{3}{3}=1$
This pair gives valid integers $(m,n)=(1,-5)$.

We have identified four valid integer pairs for $(m,n)$. We now test each of them by plugging them into the target expression, $2m-3n$, to determine its maximum value:

For $(5,-1)$:
$2\left(5\right)-3(-1)=10+3=13$

For $(-1,5)$:
$2(-1)-3\left(5\right)=-2-15=-17$

For $(-5,1)$:
$2(-5)-3\left(1\right)=-10-3=-13$

For $(1,-5)$:
$2\left(1\right)-3(-5)=2+15=17$

Comparing the outcomes (13, -17, -13, 17), the greatest value produced is 17.