Find the unit digit of expression
`(232)^(123!) × (353)^(124!) × (424)^(124!)`

Options

A. 3
B. 4
C. 6
D. 8

Answer

To find the unit digit of the expression \( (232)^{123!} \times (353)^{124!} \times (424)^{124!} \), we will focus on the unit digits of each base number raised to their respective powers.

### Step 1: Identify the unit digits of the bases
- The unit digit of \( 232 \) is \( 2 \).
- The unit digit of \( 353 \) is \( 3 \).
- The unit digit of \( 424 \) is \( 4 \).

### Step 2: Analyze the powers
We need to find the unit digits of:
- \( 2^{123!} \)
- \( 3^{124!} \)
- \( 4^{124!} \)

### Step 3: Determine the unit digit of \( 2^{123!} \)
The unit digits of powers of \( 2 \) cycle every 4:
- \( 2^1 = 2 \)
- \( 2^2 = 4 \)
- \( 2^3 = 8 \)
- \( 2^4 = 6 \)
- \( 2^5 = 2 \) (and so on...)

To find the unit digit of \( 2^{123!} \), we need to calculate \( 123! \mod 4 \):
- Since \( 123! \) contains all integers from \( 1 \) to \( 123 \), it includes \( 4 \), so \( 123! \equiv 0 \mod 4 \).
- Therefore, \( 2^{123!} \equiv 2^0 \equiv 6 \) (from the cycle).

### Step 4: Determine the unit digit of \( 3^{124!} \)
The unit digits of powers of \( 3 \) cycle every 4:
- \( 3^1 = 3 \)
- \( 3^2 = 9 \)
- \( 3^3 = 7 \)
- \( 3^4 = 1 \)
- \( 3^5 = 3 \) (and so on...)

Now, we calculate \( 124! \mod 4 \):
- Similar to \( 123! \), \( 124! \) also contains \( 4 \), so \( 124! \equiv 0 \mod 4 \).
- Therefore, \( 3^{124!} \equiv 3^0 \equiv 1 \).

### Step 5: Determine the unit digit of \( 4^{124!} \)
The unit digits of powers of \( 4 \) cycle every 2:
- \( 4^1 = 4 \)
- \( 4^2 = 6 \)
- \( 4^3 = 4 \) (and so on...)

Now, we calculate \( 124! \mod 2 \):
- Since \( 124! \) includes all even numbers, \( 124! \equiv 0 \mod 2 \).
- Therefore, \( 4^{124!} \equiv 4^0 \equiv 6 \).

### Step 6: Combine the unit digits
Now we have:
- Unit digit of \( 2^{123!} \) is \( 6 \).
- Unit digit of \( 3^{124!} \) is \( 1 \).
- Unit digit of \( 4^{124!} \) is \( 6 \).

Now, we multiply these unit digits together:
- \( 6 \times 1 \times 6 = 36 \).

### Step 7: Find the unit digit of the product
The unit digit of \( 36 \) is \( 6 \).

### Final Answer
The unit digit of the expression \( (232)^{123!} \times (353)^{124!} \times (424)^{124!} \) is \( 6 \).