The number of positive integral solutions of the equation `x_(1)x_(2)x_(3)=60` is:

Options

A. 54
B. 27
C. 81
D. none

Answer

To find the number of positive integral solutions of the equation \( x_1 x_2 x_3 = 60 \), we can follow these steps:

### Step 1: Prime Factorization of 60
First, we need to factor the number 60 into its prime factors.
\[ 60 = 2^2 \times 3^1 \times 5^1 \]

### Step 2: Express Each Variable in Terms of Prime Factors
We can express each variable \( x_1, x_2, \) and \( x_3 \) in terms of the prime factors:
\[ x_1 = 2^{p_1} \times 3^{p_2} \times 5^{p_3} \]
\[ x_2 = 2^{q_1} \times 3^{q_2} \times 5^{q_3} \]
\[ x_3 = 2^{r_1} \times 3^{r_2} \times 5^{r_3} \]

### Step 3: Set Up Equations for Each Prime Factor
From the equation \( x_1 x_2 x_3 = 60 \), we can derive the following equations based on the powers of the prime factors:
1. For the factor of 2:
\[ p_1 + q_1 + r_1 = 2 \]
2. For the factor of 3:
\[ p_2 + q_2 + r_2 = 1 \]
3. For the factor of 5:
\[ p_3 + q_3 + r_3 = 1 \]

### Step 4: Find the Number of Non-Negative Integral Solutions
We will use the "stars and bars" theorem to find the number of non-negative integral solutions for each equation.

#### For \( p_1 + q_1 + r_1 = 2 \):
Using the formula for the number of solutions:
\[ \text{Number of solutions} = \binom{n + r - 1}{r - 1} \]
where \( n = 2 \) (the total sum) and \( r = 3 \) (the number of variables):
\[ \text{Number of solutions} = \binom{2 + 3 - 1}{3 - 1} = \binom{4}{2} = 6 \]

#### For \( p_2 + q_2 + r_2 = 1 \):
\[ \text{Number of solutions} = \binom{1 + 3 - 1}{3 - 1} = \binom{3}{2} = 3 \]

#### For \( p_3 + q_3 + r_3 = 1 \):
\[ \text{Number of solutions} = \binom{1 + 3 - 1}{3 - 1} = \binom{3}{2} = 3 \]

### Step 5: Calculate the Total Number of Solutions
Now, we multiply the number of solutions for each prime factor:
\[ \text{Total number of solutions} = 6 \times 3 \times 3 = 54 \]

### Final Answer
Thus, the number of positive integral solutions of the equation \( x_1 x_2 x_3 = 60 \) is \( \boxed{54} \).

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