辅导案例-PMATH 340-Assignment 2

• June 23, 2020

PMATH 340 Number Theory, Assignment 2 Due: Tues June 23 Read Chapters 3 and 4 in the Lecture Notes, and work on the Exercises for Chapters 3 and 4 in the Practice Problems. Then solve each of the following problems. 1: (a) Find 1050 mod 91. (b) Find 2827 26 mod 25. (c) Find a positive integer k such that the number 3k ends with the digits 0001. (d) With the help of the following table of powers of 5 mod 64, solve 11×5 = 17 mod 64. k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 5k 1 5 25 61 49 53 9 45 33 37 57 29 17 21 41 13 1 −5k 63 59 39 3 15 11 55 19 31 27 7 35 47 43 23 51 63 2: (a) For n = 675, find a positive integer ` and find primes p1, p2, · · · , p` and positive integers k1, k2, · · · , k` such that Un ∼= Zp1k1 × Zp2k2 × · · · × Zp`k` . (b) For n = 125, find the number squares, the number of cubes, and the number of fourth powers in Un. (c) For n = 18900, find the universal exponent λ(n) and find the number of elements in Un of order λ(n). 3: (a) Evaluate the Legendre symbol ( 23 61 ) . (b) For n = 1111, determine whether 47 ∈ Qn. (c) For n = 400, determine the number of elements in Qn. (d) Show that for primes p > 3 we have (−3 p ) = 1 ⇐⇒ p = 1 mod 6.

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