Mathematics 228(Q1), Assignment 3
Due : Monday, February 5, 2007
Exercise 1.(10 marks)(a) If mis an odd integer, show m2≡1 mod 8.
(b) Let mbe an odd integer. Show m2n≡1 mod 2n+2 for all positive natural numbers n.
Exercise 2.(10 marks)(a) If mis a nonnegative integer, show mis congruent modulo 10 to its last digit.(eg.
27 ≡7 mod 10).
(b) Show that no perfect square has 2, 3, 7, or 8 as its last digit. (Hint : Work modulo 10.)
Exercise 3.(10 marks) Solve the following equations.
(a) 25x= 3 in Z77.
(b) 243x+ 17 = 101 in Z725.
Exercise 4.(15 marks) Let n∈N,n > 1. Let a,bbe integers and set d= (a, n).
(a) Assuming the equation [a]x= [b] has a solution in Zn, show that d|b.
(b) Conversely, assume d|b.
(i) Explain why there exists integers u,v,w,a1,b1, and n1such that
au +nv =d, a =da1, b =db1,and n=dn1.
(ii) Show that
x= [ub1+in1],0≤i≤d−1,(1)
are all solution of the equation [a]x= [b] in Zn.
Exercise 5.(10 marks) Let the notation be as in Exercise 3. We here assume d|b.
(a) Show that the solutions of the equation [a]x= [b] in Znprovided by (1) are distinct.
(b) If x= [r] is any solution of [a]x= [b] in Znthen [r] = [ub1+in1] for a suitable integer i, 0 ≤i≤
k−1. (Hint : Observe that [ar]−[aub1] = [0]. Use this to show first n1|(a1(r−ub1)), and then
deduce n1|(r−ub1).)
Exercise 6.(15 marks) Find all the solutions of the following equations. (Hint : You may find the results of
exercises 4 and 6 useful.)
(a) 36x= 78 in Z96
(b) 98x= 175 in Z245
(c) 35x= 63 in Z77 .