If a, b, and c are positive integers such that `(a-b+c)(b-c+a)(c-a+b)=15`, then what is the product of a, b and c?

Options

A. 24
B. 64
C. 42
D. Cannot be determined

Answer

To solve the problem, we need to find the positive integers \( a, b, \) and \( c \) such that:

\[ (a - b + c)(b - c + a)(c - a + b) = 15 \]

### Step 1: Factor the number 15
The number 15 can be factored in several ways using positive integers:

1. \( 1 \times 1 \times 15 \)
2. \( 1 \times 3 \times 5 \)

Since \( a, b, \) and \( c \) are positive integers, we will focus on the factorization \( 1 \times 3 \times 5 \).

### Step 2: Assign values to the factors
Let’s assign:

- \( a - b + c = 1 \)
- \( b - c + a = 3 \)
- \( c - a + b = 5 \)

### Step 3: Solve the equations
Now we will solve these equations step by step.

1. From the first equation:
\[ a - b + c = 1 \quad \text{(Equation 1)} \]

2. From the second equation:
\[ b - c + a = 3 \quad \text{(Equation 2)} \]

3. From the third equation:
\[ c - a + b = 5 \quad \text{(Equation 3)} \]

### Step 4: Rearranging the equations
Now we can rearrange these equations to express \( a, b, \) and \( c \):

From Equation 1:
\[ a + c = b + 1 \quad \text{(1')} \]

From Equation 2:
\[ a + b = c + 3 \quad \text{(2')} \]

From Equation 3:
\[ b + c = a + 5 \quad \text{(3')} \]

### Step 5: Solve for \( a, b, c \)
Now we will add the rearranged equations:

Adding (1') and (2'):
\[ (a + c) + (a + b) = (b + 1) + (c + 3) \]
This simplifies to:
\[ 2a + b + c = b + c + 4 \]
Cancelling \( b + c \) from both sides gives:
\[ 2a = 4 \implies a = 2 \]

Now substituting \( a = 2 \) into (1'):
\[ 2 + c = b + 1 \implies c = b - 1 \]

Substituting \( a = 2 \) into (2'):
\[ 2 + b = c + 3 \implies c = b - 1 \]

Substituting \( c = b - 1 \) into (3'):
\[ b + (b - 1) = 2 + 5 \implies 2b - 1 = 7 \implies 2b = 8 \implies b = 4 \]

Now substituting \( b = 4 \) back to find \( c \):
\[ c = b - 1 = 4 - 1 = 3 \]

### Final values
Thus, we have:
- \( a = 2 \)
- \( b = 4 \)
- \( c = 3 \)

### Step 6: Calculate the product
Now, we need to find the product \( a \times b \times c \):
\[ a \times b \times c = 2 \times 4 \times 3 = 24 \]

### Conclusion
The product of \( a, b, \) and \( c \) is \( 24 \).