What is the units digit of `12^(81) xx 13^(87)`?

Options

A. 6
B. 3
C. 4
D. 2

Answer

To find the units digit of \(12^{81} \times 13^{87}\), we can follow these steps:

### Step 1: Find the units digit of \(12^{81}\)
The units digit of a number raised to a power depends only on the units digit of the base. The units digit of \(12\) is \(2\). Therefore, we need to find the units digit of \(2^{81}\).

### Step 2: Identify the pattern in the units digits of powers of \(2\)
The units digits of powers of \(2\) follow a repeating cycle:
- \(2^1 = 2\) (units digit is \(2\))
- \(2^2 = 4\) (units digit is \(4\))
- \(2^3 = 8\) (units digit is \(8\))
- \(2^4 = 16\) (units digit is \(6\))
- \(2^5 = 32\) (units digit is \(2\)) - and the cycle repeats.

The cycle is: \(2, 4, 8, 6\) and it repeats every \(4\) terms.

### Step 3: Determine the position in the cycle for \(2^{81}\)
To find which units digit corresponds to \(2^{81}\), we calculate \(81 \mod 4\):
\[ 81 \div 4 = 20 \quad \text{remainder } 1 \]
Thus, \(81 \mod 4 = 1\). This means \(2^{81}\) corresponds to the first position in the cycle, which is \(2\).

### Step 4: Find the units digit of \(13^{87}\)
Next, we find the units digit of \(13^{87}\). The units digit of \(13\) is \(3\). Therefore, we need to find the units digit of \(3^{87}\).

### Step 5: Identify the pattern in the units digits of powers of \(3\)
The units digits of powers of \(3\) also follow a repeating cycle:
- \(3^1 = 3\) (units digit is \(3\))
- \(3^2 = 9\) (units digit is \(9\))
- \(3^3 = 27\) (units digit is \(7\))
- \(3^4 = 81\) (units digit is \(1\))
- \(3^5 = 243\) (units digit is \(3\)) - and the cycle repeats.

The cycle is: \(3, 9, 7, 1\) and it repeats every \(4\) terms.

### Step 6: Determine the position in the cycle for \(3^{87}\)
To find which units digit corresponds to \(3^{87}\), we calculate \(87 \mod 4\):
\[ 87 \div 4 = 21 \quad \text{remainder } 3 \]
Thus, \(87 \mod 4 = 3\). This means \(3^{87}\) corresponds to the third position in the cycle, which is \(7\).

### Step 7: Combine the units digits
Now we have:
- The units digit of \(12^{81}\) is \(2\).
- The units digit of \(13^{87}\) is \(7\).

To find the units digit of \(12^{81} \times 13^{87}\), we multiply these units digits:
\[ 2 \times 7 = 14 \]
The units digit of \(14\) is \(4\).

### Final Answer
The units digit of \(12^{81} \times 13^{87}\) is \(4\).

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