What is the unit digit in `[4523^(1632)xx2224^(1632) xx 3225^(1632)]` ?

Options

A. A. 1
B. B. 0
C. C. 4
D. D. 5

Answer

To find the unit digit of the expression \( 4523^{1632} \times 2224^{1632} \times 3225^{1632} \), we can follow these steps:

### Step 1: Identify the unit digits of the bases
- The unit digit of \( 4523 \) is \( 3 \).
- The unit digit of \( 2224 \) is \( 4 \).
- The unit digit of \( 3225 \) is \( 5 \).

### Step 2: Analyze the unit digit of each term raised to the power \( 1632 \)

#### For \( 3^{1632} \):
- The unit digits of powers of \( 3 \) cycle through \( 3, 9, 7, 1 \) (with a cycle length of 4).
- Since \( 1632 \mod 4 = 0 \), we take the unit digit corresponding to \( 3^0 \), which is \( 1 \).

#### For \( 4^{1632} \):
- The unit digits of powers of \( 4 \) cycle through \( 4, 6 \) (with a cycle length of 2).
- Since \( 1632 \mod 2 = 0 \), we take the unit digit corresponding to \( 4^0 \), which is \( 6 \).

#### For \( 5^{1632} \):
- The unit digit of any power of \( 5 \) is always \( 5 \).

### Step 3: Combine the unit digits
Now we have the unit digits:
- From \( 4523^{1632} \): \( 1 \)
- From \( 2224^{1632} \): \( 6 \)
- From \( 3225^{1632} \): \( 5 \)

### Step 4: Multiply the unit digits
Now we multiply these unit digits together:
\[ 1 \times 6 \times 5 = 30 \]

### Step 5: Find the unit digit of the result
The unit digit of \( 30 \) is \( 0 \).

### Final Answer
Thus, the unit digit in \( 4523^{1632} \times 2224^{1632} \times 3225^{1632} \) is \( 0 \).

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