### Instruction: What is the smallest positive integer $x$ that, when multiplied by $400$, produces a product that is a multiple of $576$?### Response: To find the smallest $x$ that makes $400x$ a multiple of $576$, I need to think about the prime factorizations of $400$ and $576$.
$400$ has four factors of $2$ and two factors of $5$: $400 = 2^4 \cdot 5^2$.
$576$ has six factors of $2$ and one factor of $3$: $576 = 2^6 \cdot 3$.
If $400x$ is a multiple of $576$, then it must have at least six factors of $2$ and one factor of $3$ as well.
It already has four factors of $2$, so it needs at least two more.
It does not have any factors of $3$, so it needs at least one.
Therefore, the smallest $x$ that works is $x = 2^2 \cdot 3 = 12$.
To check, $400 \cdot 12 = 4800$, and $4800$ is indeed a multiple of $576$.