A number is a multiple of $q$ and $r$ if and only if it is divisible by both $q$ and $r$.

Given Information
- $q$ and $r$ are two distinct prime numbers.
- $q>20$ and $r>20$.
- Since $q$ and $r$ are distinct prime numbers, they share no common factors other than 1 . Their Least Common Multiple (LCM) is simply their product, $q r$.

Any multiple of both $q$ and $r$ must be a multiple of their LCM, $q r$.

Analysis of Options
We check if each option contains $q$ and $r$ as factors.
A. $\(q^3 r^2\)$

$$
\(q^3 r^2=\left(q \cdot q^2\right) \cdot(r \cdot r)=q r \cdot\left(q^2 r\right)\)
$$


Since the expression contains the factor $q r$, it is divisible by $q$ and divisible by $r$.
- It is a multiple of $q$.
- It is a multiple of $r$. Result: Is a multiple of $q$ and $r$.

B. $16 q$ This expression is guaranteed to be a multiple of $q$. Is it guaranteed to be a multiple of $r$ ? Only if $r$ is a factor of $16 q$. Since $r$ is a prime number greater than $20, r$ cannot be 16 or any factor of 16 (like 2 or 4). Since $r$ is prime and distinct from $q$, $r$ is not a factor of $16 q$ unless $r=$ 16 , which is impossible because $r$ is a prime greater than 20 .

- For example, if $q=23$ and $\(r=29,16 q=16 \times 23\)$. This is not divisible by 29 . Result: Is NOT guaranteed to be a multiple of $r$.

C. $9 r$ This expression is guaranteed to be a multiple of $r$. Is it guaranteed to be a multiple of $q$ ? Only if $q$ is a factor of $9 r$. Since $q$ is a prime number greater than $20, q$ cannot be 9 or 3 (factors of 9). Since $q$ is prime and distinct from $r, q$ is not a factor of $9 r$.

- For example, if $q=23$ and $\(r=29,9 r=9 \times 29\)$. This is not divisible by 23 . Result: Is NOT guaranteed to be a multiple of $q$.
D. $52 q r$

$$
\(52 q r=52 \times(q r)\)
$$


Since the expression contains the factor $q r$, it is clearly divisible by $q$ and divisible by $r$.
- It is a multiple of $q$.
- It is a multiple of $r$. Result: Is a multiple of $q$ and $r$.

The expressions that could be multiples of $q$ and $r$ are (A) $\(q^3 r^2\)$ and (D) $\(52 q r\)$.