Competition problem — Number theory

Library / /61 of 520

Number theory Difficulty 5.5 AIME, harder Find the answer

[6.1] (i) Find all positive integers $n$ such that $2^{n}-1$ is divisible by 7;
(ii) Prove that for any positive integer $n, 2^{n}+1$ cannot be divisible by 7.

Your answer

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

Solution

None

Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.

Note: The provided instruction is a meta-instruction and not part of the text to be translated. Since the text to be translated is "None", the translation is also "None". Here is the formatted output as requested:

None

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

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