Midterm 1 - Let p be a prime, a be an integer with ( a,p )...

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MATH 115 MID-TERM 1 1. (4 pts) a) (3 pts) Given the factorisation 1728 = 2 6 × 3 3 , show that for any integer a which is relatively prime to 7 , 13 and 19, we have a 1728 1 mod 7 a 1728 1 mod 13 a 1728 1 mod 19 (hint: use Fermat’s little theorem.) b) (1 pt) Given the factorisation 1729 = 7 × 13 × 19, use part a) to show that for any a with ( a, 1729) = 1, we have a 1728 1 mod 1729 2. (6 pts) Find all the solutions to the congruence equation: x 3 - 5 x 2 + 3 0 mod 27 using the algorithm we learnt in class. State your steps. 3. (3 pts) Solve the simultaneous congruence 2 x 3 mod 7 3 x ≡ - 1 mod 11 4. (2 pts) Let a,b, m be integers with ( a,m ) = 1. Show that the solution to the congruence equation ax b mod m is given by x ba φ ( m ) - 1 mod m . (where φ is Euler’s function.) 5. (5 pts)
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Unformatted text preview: Let p be a prime, a be an integer with ( a,p ) = 1. a) (2 pts) Show that { a, 2 a, , ( p-1) a } is a reduced residue system for the modulus p . b) (2 pts) Let N k = 1 k + 2 k + + ( p-1) k . Use the results of part a) to show that a k N k N k mod p 1 2 MATH 115 MID-TERM 1 for any a with ( a,p ) = 1. Note that the result of part b) can be written as ( a k-1) N k mod p for any ( a,p ) = 1. c) (1 pt) In part b), take the numbner a to be a primitive root g mod p . Show that N k mod p whenever k is not divisible by p-1. (hint: since g is a primitive root modulo p , we have ord p ( g ) = p-1.)...
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Midterm 1 - Let p be a prime, a be an integer with ( a,p )...

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