If ` a^(3) b = abc ` = 180, a, b, c are positive integers , then the value of c is

Options

A. 110
B. 1
C. 4
D. 25

Answer

To solve the problem, we start with the equations given:

1. \( a^3 b = abc \)
2. \( abc = 180 \)

We need to find the value of \( c \) given that \( a \), \( b \), and \( c \) are positive integers.

### Step 1: Simplify the first equation
From the first equation, we can rearrange it as follows:

\[ a^3 b = abc \]

We can divide both sides by \( ab \) (where \( a \) and \( b \) are not zero):

\[ a^2 = c \]

### Step 2: Substitute \( c \) in the second equation
Now, we substitute \( c \) in the second equation:

\[ ab(a^2) = 180 \]

This simplifies to:

\[ a^3 b = 180 \]

### Step 3: Express \( b \) in terms of \( a \)
From the equation \( a^3 b = 180 \), we can express \( b \) as:

\[ b = \frac{180}{a^3} \]

### Step 4: Ensure \( b \) is a positive integer
For \( b \) to be a positive integer, \( 180 \) must be divisible by \( a^3 \). Therefore, we need to find the values of \( a \) such that \( a^3 \) is a divisor of \( 180 \).

### Step 5: Factor \( 180 \)
The prime factorization of \( 180 \) is:

\[ 180 = 2^2 \times 3^2 \times 5^1 \]

### Step 6: Find possible values of \( a \)
Now we will find the possible values of \( a \) such that \( a^3 \) is a divisor of \( 180 \):

- \( a = 1 \): \( a^3 = 1 \) (divides \( 180 \))
- \( a = 2 \): \( a^3 = 8 \) (divides \( 180 \))
- \( a = 3 \): \( a^3 = 27 \) (does not divide \( 180 \))
- \( a = 4 \): \( a^3 = 64 \) (does not divide \( 180 \))
- \( a = 5 \): \( a^3 = 125 \) (does not divide \( 180 \))

The only valid values for \( a \) are \( 1 \) and \( 2 \).

### Step 7: Calculate \( c \) for valid \( a \) values
1. If \( a = 1 \):
- \( c = a^2 = 1^2 = 1 \)
- \( b = \frac{180}{1^3} = 180 \)
- \( (a, b, c) = (1, 180, 1) \)

2. If \( a = 2 \):
- \( c = a^2 = 2^2 = 4 \)
- \( b = \frac{180}{2^3} = \frac{180}{8} = 22.5 \) (not an integer)

Thus, the only valid solution occurs when \( a = 1 \), leading to \( c = 1 \).

### Conclusion
The value of \( c \) is:

\[ \boxed{1} \]