Find the unit digit in the product `(4387)^245 xx (621)^72`

Options

A. 1
B. 2
C. 5
D. 7

Answer

To find the unit digit of the product \( (4387)^{245} \times (621)^{72} \), we can follow these steps:

### Step 1: Identify the unit digits
The unit digit of \( 4387 \) is \( 7 \) and the unit digit of \( 621 \) is \( 1 \).

### Step 2: Calculate the unit digit of \( (4387)^{245} \)
Since the unit digit is \( 7 \), we need to find the unit digit of \( 7^{245} \). The unit digits of powers of \( 7 \) follow a cyclic pattern:
- \( 7^1 = 7 \) (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 cycle is \( 7, 9, 3, 1 \) and it repeats every \( 4 \) terms.

### Step 3: Find the position in the cycle
To find the position of \( 245 \) in the cycle, we calculate \( 245 \mod 4 \):
\[ 245 \div 4 = 61 \quad \text{remainder } 1 \]
So, \( 245 \mod 4 = 1 \). This means \( 7^{245} \) has the same unit digit as \( 7^1 \), which is \( 7 \).

### Step 4: Calculate the unit digit of \( (621)^{72} \)
The unit digit of \( 621 \) is \( 1 \). Any power of \( 1 \) will always have a unit digit of \( 1 \):
\[ (1)^{72} = 1 \]

### Step 5: Combine the results
Now, we multiply the unit digits obtained from both calculations:
\[ \text{Unit digit of } (4387)^{245} \times (621)^{72} = 7 \times 1 = 7 \]

### Final Answer
The unit digit in the product \( (4387)^{245} \times (621)^{72} \) is \( 7 \).

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