## Description

**Question 1. [4 Marks]**

- Find a natural number
*n*such that 2 · 1066 + 1492 ≡*n*(mod 1776)*.*Is*n*unique? - Show that there is no
*n*^{∈ }N such that*n*≡ 3 (mod 4) and*n*≡ 5 (mod 8).

**Question 2. [4 Marks] **Identify the shaded regions, using only the union, intersection and complement symbols.

**Question 3. [4 Marks] **What is the coefficient of *x*^{5 }in the expression (−4 + 3*x*)^{12}? (Do NOT multiply −4 + 3*x *by itself 12 times.)

**Question 4. [8 Marks] **Prove or disprove that the following are equivalence relations. If you find one (or both) that is an equivalence relation, write out [−1] and .

- For
*a,b,c,d*^{∈ }Z with - For
*x,y*∈ R:*R*= {(*x,y*) : |*x*−*y*| ≤ 1}*.*