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1. Let p be a positive integer. Recall that Zp = {0, 1,...,p-1} where addition and multiplication
are mod p (i.e. p = 0 and a.b is the remainder of a.b divided by p). In this problem, all numbers
are suppose to be in Zp.
(a) Show that if p is not prime, then there is a, b 0 such that a.b = 0.
(b) Show that if p is prime then, for all a, b 0 such that a.b # 0
(c) Show that if p is prime, a ‡ a' € Zp and b E Zp, b.a‡b.a'.
(d) Show that if p is prime, for all be Zp the map aab is a bijection from Zp to Zp.
(e) Show that if p is prime, we can define a division operation in Zp, i.e. to each a, b 0 there
is a number (alb) € Zp such that b.(a/b) = a).
(f) How many solutions does the following system have in Z5. Give the solution set using
parameters.
2x12x2x3 = 2
3x1x2 + 4x3 = 3
(g) What would be the issue to apply Gauss Jordan with this system in Z6?
