My Answer
The key to this problem is realizing that when positive integers are raised to powers, the one's digit follows identifiable patterns. Note the pattern with 3 (or any number ending with 3)
3 to the zero power is 1
3 to the first power is 3
3 to the second power is 9
3 to the third power is 27 (one's digit is 7)
3 to the fourth power is 81 (one's digit is 1 again)
If you were to keep going you notice the pattern is 1, 3,9, and 7. It then repeats. Every fourth power has the one's digit as 1.
The number 4131 is divisible by 4 with a remainder of 3. Therefore, the number raised to the 4128th power would have a "1" as its one's digit. Following the pattern, the next power (4129) would have a "3" in the one's digit. The next power (4130) would have a "9" in the one's digit. Finally, raising a number ending in 3 to the 4131 power would have a "7" in the one's digit.
My answer is 7.
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