4. (10 points) Given a positive integer $A$ that can be factored 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
【Analysis】Half of $A$ is a perfect square, $\alpha-1 、 \beta 、 \gamma$ are multiples of 2; One-third of $A$ is a perfect cube, $\alpha 、 \beta-1 、 \gamma$ are multiples of 3; One-fifth of $A$ is a fifth power of some natural number, $\alpha 、 \beta 、 \gamma-1$ are multiples of 5; To find the minimum value of $\alpha+\beta+\gamma$, we need to determine the values of $\alpha 、 \beta 、 \gamma$ that satisfy the conditions, and then find the minimum value of $\alpha+\beta+\gamma$.
【Solution】Solution: Half of $A$ is a perfect square, $\alpha-1 、 \beta 、 \gamma$ are multiples of 2;
One-third of $A$ is a perfect cube, $\alpha 、 \beta-1 、 \gamma$ are multiples of 3;
One-fifth of $A$ is a fifth power of some natural number, $\alpha 、 \beta 、 \gamma-1$ are multiples of 5;
To find the minimum value of $\alpha+\beta+\gamma$, we need to determine the values of $\alpha 、 \beta 、 \gamma$ that satisfy the conditions: $3 \times 5-1$ is a multiple of 2, the minimum value of $\alpha$ is 15,
$2 \times 3-1$ is a multiple of 5, the minimum value of $\gamma$ is 6,
$2 \times 5-1$ is a multiple of 3, the minimum value of $\beta$ is 10,
Therefore, the minimum value of $\alpha+\beta+\gamma$ is: $15+6+10=31$;
Hence, the answer is: $D$.