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 x5 in the expression (−4 + 3x)12? (Do NOT multiply −4 + 3x 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}.
