当写成质因数分解形式时， $p$ 是 $2^2 \times 3 \times 5$， $q$ 是 $2^6 \times 3^3$， $r$ 是 $2^2 \times 3^2 \times 11$。 求

1. Find the cube root of $q$ from its prime factorisation.

Given $q=2^6\times 3^3$,

$\sqrt[3]{q}=\sqrt[3]{2^6\times 3^3}=2^{6\div 3}\times 3^{3\div 3}=2^2\times 3=4\times 3=12$.

1. Select the highest power of each prime factor appearing in $p$, $q$ and $r$.

$p=2^2\times3\times5$

$q=2^6\times3^3$

$r=2^2\times3^2\times11$

Highest powers: $2^6$, $3^3$, $5$ and $11$.

So the LCM is
$2^6\times3^3\times5\times11$.

1. Find the greatest common divisor of $p$, $q$ and $r$ from their prime factorisations.

Common prime factors are $2$ and $3$.

Take the smallest powers:
$2^2 \times 3 = 4 \times 3 = 12$

So the greatest number that divides $p$, $q$ and $r$ exactly is $12$.