Unclassified Questions: Difference between revisions

Revision as of 23:14, 24 March 2023

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

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

Submitted by BinouTheGuineaPig | A step-by-step solution

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

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

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

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

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

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

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

$\qquad\qquad\equiv 28496\mod811$

$\qquad\qquad\equiv 111\mod811$

@@ Line 3: / Line 3: @@
 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)} 229</math>
+<math>\textrm{(A) 111}\qquad\textrm{(B) 142}\qquad\textrm{(C) 157}\qquad\textrm{(D) 221}\qquad\textrm{(E) 229}</math>
 ===Solution===