Find the unit’s digit in the product of `37562^(156!)` and `6787^(675!)`

Options

A. 4
B. 0
C. 8
D. 6

Answer

To find the unit's digit in the product of \( 37562^{156!} \) and \( 6787^{675!} \), we can follow these steps:

### Step 1: Find the unit's digit of \( 37562^{156!} \)

1. **Identify the unit's digit of the base**: The unit's digit of \( 37562 \) is \( 2 \).
2. **Determine the unit's digit of \( 2^{156!} \)**: The unit's digits of powers of \( 2 \) follow a pattern:
- \( 2^1 = 2 \) (unit's digit is 2)
- \( 2^2 = 4 \) (unit's digit is 4)
- \( 2^3 = 8 \) (unit's digit is 8)
- \( 2^4 = 16 \) (unit's digit is 6)
- \( 2^5 = 32 \) (unit's digit is 2)

The pattern repeats every 4 terms: \( 2, 4, 8, 6 \).

3. **Find \( 156! \mod 4 \)**: Since \( 156! \) contains multiples of \( 4 \), we know \( 156! \equiv 0 \mod 4 \).

4. **Use the pattern**: Since \( 156! \equiv 0 \mod 4 \), the unit's digit of \( 2^{156!} \) corresponds to the 4th term in the pattern, which is \( 6 \).

### Step 2: Find the unit's digit of \( 6787^{675!} \)

1. **Identify the unit's digit of the base**: The unit's digit of \( 6787 \) is \( 7 \).
2. **Determine the unit's digit of \( 7^{675!} \)**: The unit's digits of powers of \( 7 \) follow a pattern:
- \( 7^1 = 7 \) (unit's digit is 7)
- \( 7^2 = 49 \) (unit's digit is 9)
- \( 7^3 = 343 \) (unit's digit is 3)
- \( 7^4 = 2401 \) (unit's digit is 1)
- \( 7^5 = 16807 \) (unit's digit is 7)

The pattern repeats every 4 terms: \( 7, 9, 3, 1 \).

3. **Find \( 675! \mod 4 \)**: Since \( 675! \) contains multiples of \( 4 \), we know \( 675! \equiv 0 \mod 4 \).

4. **Use the pattern**: Since \( 675! \equiv 0 \mod 4 \), the unit's digit of \( 7^{675!} \) corresponds to the 4th term in the pattern, which is \( 1 \).

### Step 3: Find the unit's digit of the product

1. **Multiply the unit's digits**: Now, we have the unit's digits:
- From \( 37562^{156!} \): \( 6 \)
- From \( 6787^{675!} \): \( 1 \)

2. **Calculate the unit's digit of the product**:
\[ 6 \times 1 = 6 \]

### Final Answer:
The unit's digit in the product \( 37562^{156!} \times 6787^{675!} \) is \( 6 \).

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