Find the unit digit of \(111^{111} \times 222^{222} \times 333^{333}\)
Answer: A
To find the unit digit of the expression, we only need to consider the unit digits of the bases. The problem is equivalent to finding the unit digit of the product \(1^{111} \times 2^{222} \times 3^{333}\).
Unit digit of \(1^{111}\) is always 1.
For \(2^{222}\), the power cycle of 2 is (2, 4, 8, 6), which repeats every 4 powers. We find the remainder of \(222 \div 4\), which is 2. So the unit digit is the second in the cycle, which is 4.
For \(3^{333}\), the power cycle of 3 is (3, 9, 7, 1), which repeats every 4 powers. We find the remainder of \(333 \div 4\), which is 1. So the unit digit is the first in the cycle, which is 3.
The unit digit of the entire product is the unit digit of \(1 \times 4 \times 3 = 12\), which is 2.
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