We want to find the smallest positive integer $x$ such that $400x$ is a multiple of $576$.
Since $576=2^6\cdot3^2$, we need $400x$ to have at least $6$ factors of $2$ and at least $2$ factors of $3$.
The prime factorization of $400$ is $2^4\cdot5^2$, so $x$ must have at least $2$ factors of $2$ and at least $2$ factors of $3$.
The smallest positive integer $x$ that meets these conditions is $2^2\cdot3^2=\boxed{36}$.
The answer is: 36
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
What is the smallest positive integer $x$ that, when multiplied by $400$, produces a product that is a multiple of $576$?

### Answer:
