To find the unit digit of the expression \( A = 3^n \times 6^m \), where \( n \) is the number of times 3 is multiplied and \( m \) is the number of times 6 is multiplied, we need to determine the values of \( n \) and \( m \). Based on the provided information, let’s consider the calculations step-by-step.

### Step 1: Analyzing the unit digits of the components
- The unit digit of \( 3^n \):
- The unit digits of powers of 3 cycle through \( 3, 9, 7, 1 \):
- \( 3^1 \equiv 3 \)
- \( 3^2 \equiv 9 \)
- \( 3^3 \equiv 7 \)
- \( 3^4 \equiv 1 \)
- This cycle repeats every 4 terms.

- The unit digit of \( 6^m \):
- The unit digit of any power of 6 is always \( 6 \).

### Step 2: Determine the total contributions to the unit digit
- The unit digit of \( A \) will be determined by the unit digit of \( 3^n \) multiplied by the unit digit of \( 6^m \):
\[
\text{Unit digit of } A = (\text{Unit digit of } 3^n) \times 6.
\]

### Step 3: Finding the unit digit of \( A \)
- Regardless of the value of \( n \) (mod 4), we have four possible unit digits for \( 3^n \):
- When \( n \equiv 0 \mod 4 \): unit digit is \( 1 \)
- When \( n \equiv 1 \mod 4 \): unit digit is \( 3 \)
- When \( n \equiv 2 \mod 4 \): unit digit is \( 9 \)
- When \( n \equiv 3 \mod 4 \): unit digit is \( 7 \)

Multiplying each of these by \( 6 \):
- Unit digit of \( A \) when \( n \equiv 0 \mod 4 \): \( 1 \times 6 = 6 \)
- Unit digit of \( A \) when \( n \equiv 1 \mod 4 \): \( 3 \times 6 = 18 \) → unit digit \( 8 \)
- Unit digit of \( A \) when \( n \equiv 2 \mod 4 \): \( 9 \times 6 = 54 \) → unit digit \( 4 \)
- Unit digit of \( A \) when \( n \equiv 3 \mod 4 \): \( 7 \times 6 = 42 \) → unit digit \( 2 \)

### Conclusion
The possible unit digits of \( A \) depend on \( n \):
- \( 6 \) (if \( n \equiv 0 \mod 4 \))
- \( 8 \) (if \( n \equiv 1 \mod 4 \))
- \( 4 \) (if \( n \equiv 2 \mod 4 \))
- \( 2 \) (if \( n \equiv 3 \mod 4 \))

Without knowing the specific values of \( n \) and \( m \), we cannot determine a unique unit digit. However, if \( n \) is specifically known, the unit digit can be deduced based on the above conditions.