Find the unit digit of expression
`(599)^122 – (125)^625 – (144)^124 + (236)^36 + (127)^121`

Options

A. 3
B. 4
C. 5
D. 7

Answer

To find the unit digit of the expression \( (599)^{122} - (125)^{625} - (144)^{124} + (236)^{36} + (127)^{121} \), we will calculate the unit digit of each term separately and then combine them according to the operations in the expression.

### Step 1: Find the unit digit of \( (599)^{122} \)

The unit digit of \( 599 \) is \( 9 \). We need to find the unit digit of \( 9^{122} \).

The unit digits of powers of \( 9 \) follow a pattern:
- \( 9^1 = 9 \) (unit digit is \( 9 \))
- \( 9^2 = 81 \) (unit digit is \( 1 \))
- \( 9^3 = 729 \) (unit digit is \( 9 \))
- \( 9^4 = 6561 \) (unit digit is \( 1 \))

Thus, the unit digits alternate between \( 9 \) and \( 1 \):
- For odd powers, the unit digit is \( 9 \).
- For even powers, the unit digit is \( 1 \).

Since \( 122 \) is even, the unit digit of \( (599)^{122} \) is \( 1 \).

### Step 2: Find the unit digit of \( (125)^{625} \)

The unit digit of \( 125 \) is \( 5 \). The unit digit of any power of \( 5 \) is always \( 5 \).

Thus, the unit digit of \( (125)^{625} \) is \( 5 \).

### Step 3: Find the unit digit of \( (144)^{124} \)

The unit digit of \( 144 \) is \( 4 \). The unit digits of powers of \( 4 \) follow a pattern:
- \( 4^1 = 4 \) (unit digit is \( 4 \))
- \( 4^2 = 16 \) (unit digit is \( 6 \))
- \( 4^3 = 64 \) (unit digit is \( 4 \))
- \( 4^4 = 256 \) (unit digit is \( 6 \))

The unit digits alternate between \( 4 \) and \( 6 \):
- For odd powers, the unit digit is \( 4 \).
- For even powers, the unit digit is \( 6 \).

Since \( 124 \) is even, the unit digit of \( (144)^{124} \) is \( 6 \).

### Step 4: Find the unit digit of \( (236)^{36} \)

The unit digit of \( 236 \) is \( 6 \). The unit digit of any power of \( 6 \) is always \( 6 \).

Thus, the unit digit of \( (236)^{36} \) is \( 6 \).

### Step 5: Find the unit digit of \( (127)^{121} \)

The unit digit of \( 127 \) is \( 7 \). The unit digits of powers of \( 7 \) follow a pattern:
- \( 7^1 = 7 \) (unit digit is \( 7 \))
- \( 7^2 = 49 \) (unit digit is \( 9 \))
- \( 7^3 = 343 \) (unit digit is \( 3 \))
- \( 7^4 = 2401 \) (unit digit is \( 1 \))

The unit digits repeat every \( 4 \):
- \( 7, 9, 3, 1 \)

To find the unit digit of \( 7^{121} \), we calculate \( 121 \mod 4 \):
- \( 121 \div 4 = 30 \) remainder \( 1 \).

Thus, the unit digit is the same as \( 7^1 \), which is \( 7 \).

### Step 6: Combine the unit digits

Now we combine the unit digits from each term:
- Unit digit of \( (599)^{122} \) is \( 1 \).
- Unit digit of \( (125)^{625} \) is \( 5 \).
- Unit digit of \( (144)^{124} \) is \( 6 \).
- Unit digit of \( (236)^{36} \) is \( 6 \).
- Unit digit of \( (127)^{121} \) is \( 7 \).

Now, we compute:
\[ 1 - 5 - 6 + 6 + 7 \]

Calculating step-by-step:
1. \( 1 - 5 = -4 \)
2. \( -4 - 6 = -10 \)
3. \( -10 + 6 = -4 \)
4. \( -4 + 7 = 3 \)

The unit digit of the entire expression is \( 3 \).

### Final Answer
The unit digit of the expression \( (599)^{122} - (125)^{625} - (144)^{124} + (236)^{36} + (127)^{121} \) is \( 3 \).