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 11x5 = 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.