2001 AMC 8 Problem 25

A multiple must be twice or three times a smaller permutation

Divide each answer choice by $2$ or $3$ when possible

A number in the $4000$, $5000$, or $7000$ range cannot be multiplied by $2$ or more and remain one of the given four-digit permutations. So the smaller number must start with $2$, and the larger number must be either double or triple it.

For doubles, only answer choices ending in $4$ can work. But $\frac{5724}{2}=2862$ and $\frac{7254}{2}=3627$, neither of which uses exactly the digits $2,4,5,7$.

For triples, the possible answer choices are those divisible by $3$: $7245,7425,7542$. Dividing gives $2415,2475,2514$, and only $2475$ uses exactly the required digits.

Therefore $7425=3\cdot2475$ is the unique listed multiple.

Thus, D is the correct answer.