Discussion
Find the remainder when \(5^{5^{5^{\cdots\infty}}}\) is divided by 10.
Answer: A
The remainder is 5.
The remainder on division by 10 is just the last digit. Every positive power of 5 ends in 5:
\[ 5^{1} = 5,\ 5^{2} = 25,\ 5^{3} = 125,\ \ldots \]
The tower of powers is some (huge) positive power of 5, so it too ends in 5, and dividing by 10 leaves 5. The exponent's size does not matter, which is why 'Cannot be determined' is wrong. 0 would need the number to end in 0, which needs a factor 2, and powers of 5 have none; 1 is the last-digit cycle of numbers ending in 1.
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