Unclassified Questions: Difference between revisions

Latest revision as of 18:44, 1 October 2025

This is a page created to list questions that come from an unknown source. It could be from a math contest that is not widely known, or from some source that is completely unknown.

2021 GMC 10B Problems/Problem 19

Find the remainder when $3^{18}-1$ is divided by $811$ .

$\textrm{(A) 111}\qquad\textrm{(B) 142}\qquad\textrm{(C) 157}\qquad\textrm{(D) 221}\qquad\textrm{(E) 229}$

Solution

Submitted by BinouTheGuineaPig | A step-by-step solution

$3^{18}-1=(3^6\cdot3^6\cdot3^6)-1$

$\qquad\qquad =(729\cdot729\cdot729)-1$

$\qquad\qquad\equiv (-82\cdot-82\cdot-82)-1\mod811$

$\qquad\qquad\equiv -(2^3)(41^3)-1\mod811$

$\qquad\qquad\equiv -(8)(41)(41^2)-1\mod811$

$\qquad\qquad\equiv -(328)(1681)-1\mod811$

$\qquad\qquad\equiv -(-483)(59)-1\mod811$

$\qquad\qquad\equiv 28496\mod811$

$\qquad\qquad\equiv 111\mod811$

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 ==2021 GMC 10B Problems/Problem 19==
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 <math>\qquad\qquad\equiv 111\mod811</math>
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Unclassified Questions: Difference between revisions

Latest revision as of 18:44, 1 October 2025

2021 GMC 10B Problems/Problem 19

Solution