The units digit of the expression `125^(813)xx553^(3703)xx4532^(828)` is

Options

A. 4
B. 2
C. 0
D. 5

Answer

To find the units digit of the expression \(125^{813} \times 553^{3703} \times 4532^{828}\), we will follow these steps:

### Step 1: Find the units digit of \(125^{813}\)

1. The units digit of \(125\) is \(5\).
2. The units digit of any power of \(5\) is always \(5\).
3. Therefore, the units digit of \(125^{813}\) is \(5\).

### Step 2: Find the units digit of \(553^{3703}\)

1. The units digit of \(553\) is \(3\).
2. To find the units digit of \(3^{3703}\), we need to determine the pattern of units digits for powers of \(3\):
- \(3^1 = 3\) (units digit is \(3\))
- \(3^2 = 9\) (units digit is \(9\))
- \(3^3 = 27\) (units digit is \(7\))
- \(3^4 = 81\) (units digit is \(1\))
- The pattern repeats every \(4\) numbers: \(3, 9, 7, 1\).
3. Now, we find \(3703 \mod 4\):
- \(3703 \div 4 = 925\) remainder \(3\).
4. Since the remainder is \(3\), the units digit of \(3^{3703}\) corresponds to the units digit of \(3^3\), which is \(7\).

### Step 3: Find the units digit of \(4532^{828}\)

1. The units digit of \(4532\) is \(2\).
2. To find the units digit of \(2^{828}\), we look at the pattern of units digits for powers of \(2\):
- \(2^1 = 2\) (units digit is \(2\))
- \(2^2 = 4\) (units digit is \(4\))
- \(2^3 = 8\) (units digit is \(8\))
- \(2^4 = 16\) (units digit is \(6\))
- The pattern repeats every \(4\) numbers: \(2, 4, 8, 6\).
3. Now, we find \(828 \mod 4\):
- \(828 \div 4 = 207\) remainder \(0\).
4. Since the remainder is \(0\), the units digit of \(2^{828}\) corresponds to the units digit of \(2^4\), which is \(6\).

### Step 4: Multiply the units digits

Now we have the units digits from each part:
- Units digit of \(125^{813}\) is \(5\).
- Units digit of \(553^{3703}\) is \(7\).
- Units digit of \(4532^{828}\) is \(6\).

We need to multiply these units digits together:
\[ 5 \times 7 = 35 \quad (\text{units digit is } 5) \]
\[ 5 \times 6 = 30 \quad (\text{units digit is } 0) \]

### Final Answer

The units digit of the expression \(125^{813} \times 553^{3703} \times 4532^{828}\) is \(0\).

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