1. Let t be an arbitrary element of mod 7, and let x=5+3t, y=2t, and z=t, all inmod 7. Answer the following questions, showing all working, and noting that all calculations must be performed inmod 7.

(a)Lett= 0, and compute [x, y, z].

(b)Lett= 3, and compute [x, y, z].

(c)Either findtmod 7such that [x, y, z] = [6,3,5], or prove that this is impossible.

(d)Consider the following system of linear equations overmod 7:

3x+2y+z=1

2x+y+6z=3

4x+4y+z=6.

Show that [x, y, z] = [5 + 3t,2t, t] is a solution to this system, for alltmod 7.