CAT 2025 Slot 2 QA Question 22: If $m$ and $n$ are integers such that $(m + 2n)(2m + n) = 27$, then the maximum possible value of $2m - 3n$ is

Question

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

Accepted Answer

Let $u = m + 2n$ and $v = 2m + n$. Then $uv = 27$.

$u + v = 3(m+n)$

So $u + v$ must be divisible by $3$. Among the ordered factor pairs of $27$, the relevant pairs are
$(3,9), (9,3), (-3,-9), (-9,-3)$

Solving the two linear equations in terms of $u$ and $v$,
$m = \dfrac{2 v - u}{3}$
$n = \dfrac{2

If mmm and nnn are integers such that (m+2n)(2m+n)=27(m + 2n)(2m + n) = 27(m+2n)(2m+n)=27, then the maximum possible value of 2m−3n2m - 3n2m−3n is