Find the unit digit of (4137)754

Step-by-Step Solution

To find the unit digit of (4137)^754, we only need to consider the unit digit of the base, which is 7, and raise it to the given power, 754.

1. Understand the Cyclicity: The powers of 7 follow a repeating cycle of four for their unit digits:

   • 7^1 = 7 (unit digit 7)

   • 7^2 = 49 (unit digit 9)

   • 7^3 = 343 (unit digit 3)

   • 7^4 = 2401 (unit digit 1)

   • The cycle is {7, 9, 3, 1}.

2. Determine the position in the cycle: Divide the exponent 754 by the cycle length of 4:

   • 754 / 4 = 188 with a remainder of 2.

3. Identify the unit digit: A remainder of 2 means the unit digit will be the second number in our cycle {7, 9, 3, 1}, which is 9.