If `A^(**)B` us the greatest common factor of A and B, `A$B` is defined as the least common multiple of A and B, and `AnnB` is defined as equal to `(A^(**)B) $ (A $ B)`, then what is the value of `12nn15`?

Options

A. 42
B. 45
C. 48
D. 60

Answer

To solve the problem of finding the value of \( 12 \text{nn} 15 \), we need to follow the definitions provided in the question step by step.

### Step 1: Understand the Definitions
- \( A^{**}B \) is the greatest common factor (GCF) of \( A \) and \( B \).
- \( A\$B \) is the least common multiple (LCM) of \( A \) and \( B \).
- \( A \cap B \) is defined as \( (A^{**}B) \$ (A\$B) \).

### Step 2: Substitute the Values
We need to find \( 12 \text{nn} 15 \), which translates to \( 12 \cap 15 \). According to the definition:
\[ 12 \cap 15 = (12^{**}15) \$ (12\$15) \]

### Step 3: Calculate \( 12^{**}15 \) (GCF)
To find \( 12^{**}15 \), we need to determine the GCF of 12 and 15.
- The factors of 12 are \( 2^2 \times 3 \).
- The factors of 15 are \( 3 \times 5 \).
- The common factor is \( 3 \).

Thus,
\[ 12^{**}15 = 3 \]

### Step 4: Calculate \( 12\$15 \) (LCM)
Next, we find \( 12\$15 \), which is the LCM of 12 and 15.
- The multiples of 12 are \( 12, 24, 36, 48, 60, \ldots \).
- The multiples of 15 are \( 15, 30, 45, 60, 75, \ldots \).
- The least common multiple is \( 60 \).

Thus,
\[ 12\$15 = 60 \]

### Step 5: Substitute Back into the Equation
Now we substitute back into the equation for \( 12 \cap 15 \):
\[ 12 \cap 15 = (12^{**}15) \$ (12\$15) = 3 \$ 60 \]

### Step 6: Calculate \( 3\$60 \) (LCM)
Finally, we need to find \( 3\$60 \), which is the LCM of 3 and 60.
- The multiples of 3 are \( 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, \ldots \).
- The multiples of 60 are \( 60, 120, 180, \ldots \).
- The least common multiple is \( 60 \).

Thus,
\[ 3\$60 = 60 \]

### Final Answer
Therefore, the value of \( 12 \text{nn} 15 \) is:
\[ \boxed{60} \]