Difference between revisions of "Unclassified Questions"
(→2021 GMC 10B Problems/Problem 19) |
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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)} 111\qquad\textrm{(B)} 142\qquad\textrm{(C)} 157\qquad\textrm{(D)} 221\qquad\textrm{(E)} </ | + | <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> |
Revision as of 23:04, 24 March 2023
2021 GMC 10B Problems/Problem 19
Find the remainder when 
is divided by 
.
Solution
Submitted by BinouTheGuineaPig | A step-by-step solution
