If the main set is Z, the integers, consider working with the four following sets: Q1 = {n is in Z | n mod 4 = 1} Q2 = {n is in Z | n mod 4 = 2} Q3 = {n is in Z | n mod 4 = 3} Q4 = {n is in Z | n mod 4 = 0} For each set, give two positive integers and one negative integer from the set. Then explain why these 4 sets are a partition of Z.

Show that af +...+ a, = a1 + ...+ ap(mod p).

In the following question: " Let a, b ∈ Z. Show that if a ≡ 5 (mod 6) and b ≡ 3 (mod 4), then 4a + 6b ≡ 6 (mod 8)." why do we ignore the -6 when substituting the variables of a and b, if we expand "4a + 6b ≡ 6 (mod 8)" we get (4a+6b)-6=8y, when we substitute we have to end up with the 8y right? Im just confused why the -6 is ignored?

NOTE: This question has a typo in it.  For B, the equation should be x^3 - 3x^2 +2x is congruent to 0 mod 5.