Competition problem — Number theory

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Number theory Difficulty 5.5 AIME, harder Prove it United States

Problem:

Pirate ships Somy and Lia are having a tough time. At the end of the year, they are both one pillage short of the minimum required for maintaining membership in the Pirate Guild, so they decide to pillage each other to bring their counts up. Somy by tradition only pillages $28 \cdot 3^{k}$ coins for integers $k$ , and Lia by tradition only pillages $82 \cdot 3^{j}$ coins for integers $j$ . Note that each pillage can have a different $k$ or $j$ . Somy and Lia work out a system where Somy pillages Lia $n$ times, Lia pillages Somy $n$ times, and after both sets of pillages Somy and Lia are financially even.
What is the smallest $n$ can be?

Solution

Solution:

Answer: 2

Clearly, $n=1$ cannot be achieved, because $28 \cdot 3^{k}$ is never a multiple of $82$ . However, two pillages is enough: Somy pillages $28$ and $28 \cdot 81$ from Lia, and Lia pillages $81$ and $81 \cdot 27$ from Somy. As is easily checked, both pillage $28 \cdot 82$ .

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