Competition problem — Number theory

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Number theory Difficulty 4.9 AIME Find the answer

A sequence is defined by $a_{0}=1$ and $a_{n}=2^{a_{n-1}}$ for $n \geq 1$ . What is the last digit (in base 10) of $a_{15}$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

6. Certainly $a_{13} \geq 2$ , so $a_{14}$ is divisible by $2^{2}=4$ . Writing $a_{14}=4 k$ , we have $a_{15}=2^{4 k}=16^{k}$ . But every power of 16 ends in 6, so this is the answer.

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