Find out the unit digit in the following expression -
`(3694)^(1793) xx (615)^(317) xx (841)^(941)`

Options

A. 5
B. 3
C. 0
D. 4

Answer

To find the unit digit of the expression \( (3694)^{1793} \times (615)^{317} \times (841)^{941} \), we can focus on the unit digits of each base number and their respective powers.

### Step 1: Identify the unit digits of the bases
- The unit digit of \( 3694 \) is \( 4 \).
- The unit digit of \( 615 \) is \( 5 \).
- The unit digit of \( 841 \) is \( 1 \).

### Step 2: Analyze the unit digit of each term
1. **For \( (3694)^{1793} \)**:
- The unit digit is \( 4 \).
- The powers of \( 4 \) cycle through \( 4, 6 \) (i.e., \( 4^1 = 4, 4^2 = 16, 4^3 = 64, 4^4 = 256, \ldots \)).
- The pattern repeats every 2 terms:
- If the exponent is odd, the unit digit is \( 4 \).
- If the exponent is even, the unit digit is \( 6 \).
- Since \( 1793 \) is odd, the unit digit of \( (3694)^{1793} \) is \( 4 \).

2. **For \( (615)^{317} \)**:
- The unit digit is \( 5 \).
- Any power of \( 5 \) has a unit digit of \( 5 \).
- Therefore, the unit digit of \( (615)^{317} \) is \( 5 \).

3. **For \( (841)^{941} \)**:
- The unit digit is \( 1 \).
- Any power of \( 1 \) has a unit digit of \( 1 \).
- Therefore, the unit digit of \( (841)^{941} \) is \( 1 \).

### Step 3: Multiply the unit digits obtained
Now we multiply the unit digits from each term:
- The unit digit of \( (3694)^{1793} \) is \( 4 \).
- The unit digit of \( (615)^{317} \) is \( 5 \).
- The unit digit of \( (841)^{941} \) is \( 1 \).

So, we need to calculate:
\[ 4 \times 5 \times 1 \]

Calculating step-by-step:
1. \( 4 \times 5 = 20 \) (unit digit is \( 0 \))
2. \( 0 \times 1 = 0 \) (unit digit remains \( 0 \))

### Final Result
The unit digit of the entire expression \( (3694)^{1793} \times (615)^{317} \times (841)^{941} \) is \( 0 \).

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