Competition problem — Number theory

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Number theory Difficulty 4.7 AIME Prove it United States

Problem:
Determine the largest integer $n$ such that $7^{2048}-1$ is divisible by $2^{n}$ .

Solution

Solution:
Answer: 14. We have
$7^{2048}-1=(7-1)(7+1)\left(7^{2}+1\right)\left(7^{4}+1\right) \cdots\left(7^{1024}+1\right)$
In the expansion, the eleven terms other than $7+1$ are divisible by $2$ exactly once, as can be checked easily with modulo $4$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.