Competition problem — Number theory

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Number theory Difficulty 4.5 AIME Find the answer

8. (HUN 1) For a given positive integer $k$ denote the square of the sum of its digits by $f_{1}(k)$ and let $f_{n+1}(k)=f_{1}\left(f_{n}(k)\right)$ . Determine the value of $f_{1991}\left(2^{1990}\right)$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

8. Since $2^{1990} > 0$ . Thus $f_{3}\left(2^{1990}\right)=r^{2}$ where $r > 1$ . Hence $f_{1991}\left(2^{1990}\right)=256$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.