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Midterm 2 Solutions

Midterm 2 Solutions - Midterm the Second Solutions 1a Find...

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Unformatted text preview: Midterm the Second: Solutions 1a. Find the number of solutions to the congruence x 13 ≡ 2 mod 53 , if any. Notice that (13 , 52) = 13. Using theorem 2.37, we check: 2 (53- 1) / (13 , 53- 1) = 2 52 / 13 = 2 4 = 16 6≡ 1 mod 53 and we see that there are no solutions. 1b. Find the number of solutions to the congruence x 20 ≡ 13 mod 61 , if any. Notice that (20 , 60) = 20. Using theorem 2.37, we check: 13 (61- 1) / (20 , 61- 1) = 13 60 / 20 = 13 3 = 2197 ≡ 1 mod 61 and we see there are 20 solutions. 1c. Let p be a prime, and p ≡ 2 mod 3 . Show that, for an integer a such that ( a,p ) = 1 , the congruence x 3 ≡ a mod p always has a unique solution. Notice that (3 ,p- 1) = (3 , 3 k +2- 1) = (3 , 3 k +1) = 1. Now, again applying theorem 2.37, we have: a ( p- 1) / 1 = a p- 1 ≡ 1 mod p so x 3 ≡ a mod p always has a solution, and as (3 ,p- 1) = 1, we know that there is only one solution, thus it is unique....
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Midterm 2 Solutions - Midterm the Second Solutions 1a Find...

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