1. Feb 27, 2010 #1

   scottstapp

   

   1. The problem statement, all variables and given/known data
   
   Reduce 3⁴⁵⁶⁷ modulo 19.
   
   2. Relevant equations
   
   
   
   3. The attempt at a solution
   I approached by first reducing 4567 modulo 18.
   
   I got the following: 3⁴⁵⁶⁷= 3^18*253+13=(3¹⁸)²⁵³*3¹³ congruent to 1²⁵³1594323 congruent to 14 mod 19
   
   Is this the correct approach? I am not sure where 14 came from in the end I just guess and checked until I found a value that worked which was 14. Any explanations on how to correctly find 14?

    

   scottstapp, Feb 27, 2010
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3. Feb 27, 2010 #2

   Tinyboss

   

   You were exactly right to use Fermat's Little Theorem to show that 3⁴⁵⁶⁷ = 3¹³ mod 19. From there I'd use binary exponentiation: 3¹³ = 3⁸⁺⁴⁺¹ = 3⁸3⁴*3.
   
   So mod 19, 3⁴ = 5, so then 3⁸ = 3⁴3⁴ = 25 mod 19 = 6 mod 19.
   
   Put it together: 3⁸3⁴*3 = 6*5*3 mod 19 = 90 mod 19 = 14.

    

   Tinyboss, Feb 27, 2010

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