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Practice Final Exam

# Practice Final Exam - -23(thus this includes both the...

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MATH 115 PRACTICE FINAL EXAM 1. Let a 2 , k 1 be positive integers. Put n = a k - 1 It’s clear that gcd( a, n ) = 1. a) (2 pts) Prove that the order of a mod n is equal to k , i.e. k is the smallest positive integer m , such that a m 1 mod n (Hint: a m - 1 < a k - 1 for m < k .) b) (2 pts) Hence show that k divides φ ( a k - 1), where φ is Euler’s function. 2. Let p be a prime, with p 3 mod 4, and a be an integer with gcd( a, p ) = 1. a) (2 pts) Show that the congruence x 4 a mod p is solvable, if and only if, a p - 1 2 1 mod p (Hint: generalized Euler’s criterion.) b) (3 pts) Hence determine whether x 4 31 mod 1999 is solvable, given that 1999 is a prime congruent to 3 mod 4. 3. Let f ( X, Y ) = 15 X 2 + 21 XY + 8 Y 2 a) (1 pt) Compute the discriminant of f . b) (4 pts) Find the reduced form equivalent to f . 4. (5 pts) Find all the reduced, positive definite, integral binary quadratic forms of

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Unformatted text preview: -23 (thus this includes both the primitite and non-primitive ones). 5. (5 pts) Find the set of all primes p 6 = 2 , 5, for which x 2 â‰¡ 10 mod p is solvable. 6. (6 pts) Compute the quadratic irrationality represented by the periodic contin-ued fraction h 2 , 5 i and h 3 , 4 i 7. (6 pts) Compute the continued fraction expansion of âˆš 11 and âˆš 30. 1 2 MATH 115 PRACTICE FINAL EXAM 8. Let p be a prime of the form p = a 2 + b 2 , with a , odd, b even. Note that in this case, p â‰¡ 1 mod 4. a) (1 pts) Show that gcd( a,b ) = 1. b) (2 pts) Show that: Â± p a Â² = 1 c) (1 pts) Hence use the quadratic reciprocity law for Jacobi symbols, together with the results of part b), to show Â± a p Â² = 1...
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