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| + | 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. | ||
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==2021 GMC 10B Problems/Problem 19== | ==2021 GMC 10B Problems/Problem 19== | ||
Find the remainder when <math>3^{18}-1</math> is divided by <math>811</math>. | Find the remainder when <math>3^{18}-1</math> is divided by <math>811</math>. | ||
| − | <math>\textrm{(A)} | + | <math>\textrm{(A) 111}\qquad\textrm{(B) 142}\qquad\textrm{(C) 157}\qquad\textrm{(D) 221}\qquad\textrm{(E) 229}</math> |
===Solution=== | ===Solution=== | ||
'''Submitted by BinouTheGuineaPig''' | ''A step-by-step solution'' | '''Submitted by BinouTheGuineaPig''' | ''A step-by-step solution'' | ||
| − | < | + | <math>3^{18}-1=(3^6\cdot3^6\cdot3^6)-1</math> |
| − | < | + | <math>\qquad\qquad =(729\cdot729\cdot729)-1</math> |
| − | < | + | <math>\qquad\qquad\equiv (-82\cdot-82\cdot-82)-1\mod811</math> |
| − | < | + | <math>\qquad\qquad\equiv -(2^3)(41^3)-1\mod811</math> |
| − | < | + | <math>\qquad\qquad\equiv -(8)(41)(41^2)-1\mod811</math> |
| − | < | + | <math>\qquad\qquad\equiv -(328)(1681)-1\mod811</math> |
| − | < | + | <math>\qquad\qquad\equiv -(-483)(59)-1\mod811</math> |
| − | < | + | <math>\qquad\qquad\equiv 28496\mod811</math> |
| − | < | + | <math>\qquad\qquad\equiv 111\mod811</math> |
Latest revision as of 13:35, 26 March 2023
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 
is divided by 
.
Solution
Submitted by BinouTheGuineaPig | A step-by-step solution
