Two integers, A and B, can be written as products of prime factors. A = $3^{m+2}\times7$
B = $q\times3^{n}\times7$ The lowest common multiple (LCM) of A and B is $3^3\times5\times7$.

1. Use the LCM exponent rule for the prime factor $3$ to find $m$.

In $A=3^{m+2}\times 7$, the exponent of $3$ is $m+2$.

The LCM is $3^3\times 5\times 7$, so the exponent of $3$ in the LCM is $3$.

Since the LCM takes the greatest exponent of each prime factor,
$m+2=3$
$m=1$

1. Identify the missing prime factor contributed by $q$ in the LCM.

The LCM is $3^3\times5\times7$, while $B=q\times3^n\times7$.

Since $q$ is a prime factor in $B$ and the LCM contains the prime factor $5$, we have

$q=5$.

1. Substitute the earlier values of $m$ and $q$, then choose the common prime factors of $A$ and $B$ with the smaller exponents.

From part (a), $m=1$ and $q=5$.

So
$A=3^{1+2}\times 7=3^3\times 7$

and
$B=5\times 3^n\times 7$.

Since the LCM has $3^3$, and $A$ already contains $3^3$, $n\leq 3$. From the given form and the LCM, the common prime factors definitely shared by $A$ and $B$ are $3$ and $7$, with the smaller power of $3$ being $3^1$.

Hence
$\text{HCF}=3\times 7=21.$