Find the unit's digit in the product `(2467)^(153)xx(8765)^(72)`.

Options

A. a) 5
B. b) 7
C. c) 8
D. d) 9

Answer

To find the unit's digit in the product \( (2467)^{153} \times (8765)^{72} \), we can follow these steps:

### Step 1: Identify the unit digits
First, we need to identify the unit digits of the numbers involved:
- The unit digit of \( 2467 \) is \( 7 \).
- The unit digit of \( 8765 \) is \( 5 \).

### Step 2: Calculate the unit digit of \( (2467)^{153} \)
Next, we need to find the unit digit of \( 7^{153} \). To do this, we can observe the pattern in the unit digits of the powers of \( 7 \):
- \( 7^1 = 7 \) (unit digit is \( 7 \))
- \( 7^2 = 49 \) (unit digit is \( 9 \))
- \( 7^3 = 343 \) (unit digit is \( 3 \))
- \( 7^4 = 2401 \) (unit digit is \( 1 \))

The unit digits repeat every 4 powers: \( 7, 9, 3, 1 \).

Now, we need to find \( 153 \mod 4 \):
\[ 153 \div 4 = 38 \quad \text{remainder } 1 \]
So, \( 153 \mod 4 = 1 \).

This means the unit digit of \( 7^{153} \) corresponds to the unit digit of \( 7^1 \), which is \( 7 \).

### Step 3: Calculate the unit digit of \( (8765)^{72} \)
Now, we find the unit digit of \( 5^{72} \). The unit digit of any power of \( 5 \) is always \( 5 \):
- \( 5^1 = 5 \)
- \( 5^2 = 25 \)
- \( 5^3 = 125 \)
- ...

Thus, the unit digit of \( 5^{72} \) is \( 5 \).

### Step 4: Multiply the unit digits
Now, we multiply the unit digits we found:
\[ \text{Unit digit of } (2467)^{153} = 7 \]
\[ \text{Unit digit of } (8765)^{72} = 5 \]
Now, we calculate the unit digit of \( 7 \times 5 \):
\[ 7 \times 5 = 35 \]
The unit digit of \( 35 \) is \( 5 \).

### Final Answer
Thus, the unit's digit in the product \( (2467)^{153} \times (8765)^{72} \) is \( 5 \).

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