# These Two Are My Homework Questions Thanks A Lot Question 41

Question 41:
(a) Use Fermat’s Little Theorem to show that, for every prime p other than 2 or 5, there is some positive integer r for which p|(10^r -1) .
(b) Is it true that, for all integers n, other than multiples of 2 and 5, there is some positive integer r for which n|(10^r -1) .
(c)  What is the relationship between these questions and decimal expansions?
Question 42:
Let gcd(10, n)=1 , and let r be the smallest positive integer for which 10^r congruent to 1 (modulo n)
(a)  Prove that 1/n has a recurring decimal expansion with period r.
(b) If n is prime, prove that r|(n-1).
(c)  Find the periods of 1/13, 1/17, 2/31, and 1/47.

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