In the setting of Banach lattices, conditions for the factorization of positive operators are provided. A map \(T\) from a Banach space \( X \) to a Köthe function space \(F(\mu)\) (a Köthe function space is a Banach space of \(\mu\)-measurable functions which is solid, contains the characteristic functions of subsets of finite measure and satisfies the Fatou property) is said to factorize strongly through \(L_r(\mu)\) if \(T\) can be written as \(M_g\circ R\), where \(R\) is an operator from \(X\) into \(L_r(\mu)\) and \(M_g\) is a multiplication operator from \(L_r(\mu)\) to \(F(\mu)\). A main result in the first part of the paper states that for \(1