`(121)^(111)xx(1287)^(53)` find Unit digit .

Answer

To find the unit digit of the expression \( (121)^{111} \times (1287)^{53} \), we can follow these steps:

### Step 1: Identify the unit digits of the base numbers
- The unit digit of \( 121 \) is \( 1 \).
- The unit digit of \( 1287 \) is \( 7 \).

### Step 2: Determine the unit digit of \( (121)^{111} \)
- Since the unit digit of \( 121 \) is \( 1 \), any power of \( 1 \) will also have a unit digit of \( 1 \).
- Therefore, the unit digit of \( (121)^{111} \) is \( 1 \).

### Step 3: Determine the unit digit of \( (1287)^{53} \)
- The unit digit of \( 1287 \) is \( 7 \).
- The unit digits of powers of \( 7 \) follow a cycle:
- \( 7^1 = 7 \) (unit digit = 7)
- \( 7^2 = 49 \) (unit digit = 9)
- \( 7^3 = 343 \) (unit digit = 3)
- \( 7^4 = 2401 \) (unit digit = 1)
- The cycle repeats every 4 terms: \( 7, 9, 3, 1 \).

### Step 4: Find the remainder of \( 53 \) when divided by \( 4 \)
- \( 53 \div 4 = 13 \) remainder \( 1 \).
- This means \( 53 \equiv 1 \mod 4 \).

### Step 5: Determine the unit digit of \( (1287)^{53} \)
- Since \( 53 \equiv 1 \mod 4 \), the unit digit of \( (1287)^{53} \) corresponds to the unit digit of \( 7^1 \), which is \( 7 \).

### Step 6: Multiply the unit digits obtained
- Now we have:
- Unit digit of \( (121)^{111} = 1 \)
- Unit digit of \( (1287)^{53} = 7 \)
- Therefore, the unit digit of the entire expression \( (121)^{111} \times (1287)^{53} \) is:
\[ 1 \times 7 = 7 \]

### Conclusion
The unit digit of \( (121)^{111} \times (1287)^{53} \) is \( 7 \).