4. Given a positive integer $A$ that can be decomposed into prime factors as $A=2^{\alpha} \times 3^{\beta} \times 5^{\gamma}$, where $\alpha, \beta, \gamma$ are natural numbers. If half of $A$ is a perfect square, one-third of $A$ is a perfect cube, and one-fifth of $A$ is a fifth power of some natural number, then the minimum value of $\alpha+\beta+\gamma$ is ().
(A) 10
(B) 17
(C) 23
(D) 31
Answer: D.
According to “half of $A$ is a perfect square,” we can know that $(\alpha-1)$, $\beta$, and $\gamma$ are all multiples of 2. According to “one-third of $A$ is a perfect cube,” we can know that $\alpha$, $(\beta-1)$, and $\gamma$ are all multiples of 3. According to “one-fifth of $A$ is a fifth power of some natural number,” we can know that $\alpha$, $\beta$, and $(\gamma-1)$ are all multiples of 5. The smallest value of $\alpha$ that satisfies all three conditions is exactly $[3,5]=15$; the smallest value of $\beta$ is exactly $[2,5]=10$; the smallest value of $\gamma$ is exactly $[2,3]=6$. Therefore, the smallest value of $\alpha+\beta+\gamma$ is $15+10+6=31$.