Unclassified Questions: Difference between revisions
Created page with "==2021 GMC 10B Problems/Problem 19== Find the remainder when <math>3^{18}-1</math> is divided by <math>811</math>. <math>(A) 111\qquad(B) 142\qquad(C) 157\qquad(D) 221\qquad..." |
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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>(A) 111\qquad(B) 142\qquad(C) 157\qquad(D) 221\qquad(E) 229< | <math>\textrm{(A)} 111\qquad\textrm{(B)} 142\qquad\textrm{(C)} 157\qquad\textrm{(D)} 221\qquad\textrm{(E)} </math>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>3^{18}-1=(3^6\cdot3^6\cdot3^6)-1<math> | ||
<math>\qquad\qquad =(729\cdot729\cdot729)-1< | </math>\qquad\qquad =(729\cdot729\cdot729)-1<math> | ||
<math>\qquad\qquad\equiv (-82\cdot-82\cdot-82)-1\mod811< | </math>\qquad\qquad\equiv (-82\cdot-82\cdot-82)-1\mod811<math> | ||
<math>\qquad\qquad\equiv -(2^3)(41^3)-1\mod811< | </math>\qquad\qquad\equiv -(2^3)(41^3)-1\mod811<math> | ||
<math>\qquad\qquad\equiv -(8)(41)(41^2)-1\mod811< | </math>\qquad\qquad\equiv -(8)(41)(41^2)-1\mod811<math> | ||
<math>\qquad\qquad\equiv -(328)(1681)-1\mod811< | </math>\qquad\qquad\equiv -(328)(1681)-1\mod811<math> | ||
<math>\qquad\qquad\equiv -(-483)(59)-1\mod811< | </math>\qquad\qquad\equiv -(-483)(59)-1\mod811<math> | ||
<math>\qquad\qquad\equiv 28496\mod811< | </math>\qquad\qquad\equiv 28496\mod811<math> | ||
<math>\qquad\qquad\equiv 111\mod811 | </math>\qquad\qquad\equiv 111\mod811$ | ||
Revision as of 23:04, 24 March 2023
2021 GMC 10B Problems/Problem 19
Find the remainder when 
is divided by 
.
229
3^{18}-1=(3^6\cdot3^6\cdot3^6)-1$$ (Error compiling LaTeX. Unknown error_msg)\qquad\qquad =(729\cdot729\cdot729)-1$$ (Error compiling LaTeX. Unknown error_msg)\qquad\qquad\equiv (-82\cdot-82\cdot-82)-1\mod811$$ (Error compiling LaTeX. Unknown error_msg)\qquad\qquad\equiv -(2^3)(41^3)-1\mod811$$ (Error compiling LaTeX. Unknown error_msg)\qquad\qquad\equiv -(8)(41)(41^2)-1\mod811$$ (Error compiling LaTeX. Unknown error_msg)\qquad\qquad\equiv -(328)(1681)-1\mod811$$ (Error compiling LaTeX. Unknown error_msg)\qquad\qquad\equiv -(-483)(59)-1\mod811$$ (Error compiling LaTeX. Unknown error_msg)\qquad\qquad\equiv 28496\mod811$$ (Error compiling LaTeX. Unknown error_msg)\qquad\qquad\equiv 111\mod811$
