The unit's digit of the number `(3127)^(173)` is

Options

A. 1
B. 7
C. 9
D. 3

Answer

To find the unit's digit of the number \( (3127)^{173} \), we can follow these steps:

### Step 1: Identify the unit's digit of the base number
The first step is to determine the unit's digit of the base number, which is 3127. The unit's digit is the last digit of the number, which is 7.

### Step 2: Use the property of powers for unit's digits
To find the unit's digit of \( (3127)^{173} \), we only need to consider the unit's digit of the base, which is 7.

### Step 3: Find the pattern in the powers of the unit's digit
Next, we look for a pattern in the unit's digits of the powers of 7:
- \( 7^1 = 7 \) (unit's digit is 7)
- \( 7^2 = 49 \) (unit's digit is 9)
- \( 7^3 = 343 \) (unit's digit is 3)
- \( 7^4 = 2401 \) (unit's digit is 1)
- \( 7^5 = 16807 \) (unit's digit is 7)

We can see that the unit's digits repeat every 4 powers: 7, 9, 3, 1.

### Step 4: Determine the relevant power
To find out which unit's digit corresponds to \( 7^{173} \), we need to find the remainder when 173 is divided by 4 (since the pattern repeats every 4):
- \( 173 \div 4 = 43 \) remainder \( 1 \)

### Step 5: Identify the unit's digit based on the remainder
The remainder is 1, which means we look at the first number in our pattern (7, 9, 3, 1). The first number corresponds to \( 7^1 \), which has a unit's digit of 7.

### Conclusion
Therefore, the unit's digit of \( (3127)^{173} \) is **7**.