4. Given that the positive integer $A$ can be decomposed into prime factors as $A=2^{\alpha} \times 3^{\beta} \times 5^{\gamma}$, where $a 、 \beta 、 \gamma$ 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 $a +\beta+\gamma$ is ( ).
A. 10
B. 17
C. 23
D. 31
Analysis: Number Theory
$A$'s one half is a perfect square, $\alpha-1, \beta, \gamma$ are multiples of 2;
$A$'s one third is a perfect cube, $a, \beta-1, \gamma$ are multiples of 3;
$A$'s one fifth is a fifth power of some natural number, $\alpha, \beta, \gamma-1$ are multiples of 5;
To minimize $a+\beta+\gamma$, find the values of $a, \beta, \gamma$ that satisfy the conditions. $3 \times 5-1$ is a multiple of 2, the minimum value of $a$ 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, the minimum value of $\alpha+\beta+\gamma$ is $15+6+10=31$. The answer is $D$.