The matrices A ∈ M3×4 (Z₅) and B ∈ M3×1 (Z₅) are given such that

A=

1	2	3	4
2	2	4	0
3	4	2	4

B=

b

1

a

-Determine, for which a, b ∈ Z₅ the rank of the matrix A|B is the largest.

-Find the solutions of the system of equations AX = B for such a, b ∈ Z₅, for which it exists and the sum a+₅ b is the smallest.

-State, with justification, the number of all solutions of the system of equations AX = B.