Use Of Fermat’s Little Theorem

Use Fermat’s Little Theorem 3.42 to show that, for every prime p other than 2 or 5, there is some positive integer r for which pl(10′ — 1).

Is it true that, for all integers n, other than multiples of 2 and 5, there is some positive integer r for which n1(10r — 1)?

What is the relationship between these questions and decimal expan-sions? 42. Let gcd(10,n) = 1, and let r be the smallest positive integer for which lOr 1 (mod n).

Prove that 1/n has a recurring decimal expansion with period r. (b) If n is prime, prove that rl(n — 1). (c) Find the periods of 1/13, 1/17, 2/31 and 1/47.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sample Solution

ACED ESSAYS