We give some new formulas about factorizations of K-k-Schur functions g^(k)_λ, analogous to the k-rectangle factorization formula s^(k)_{Rt u λ} = s^(k)_Rt s^(k)_λ of k-Schur functions, where λ is any k-bounded partition and R_t denotes the partition (t^k+1-t) called a k-rectangle. Although a formula of the same form does not hold for K-k-Schur functions, we can prove that g^(k)_Rt divides g^(k)_{Rt u λ}, and in fact more generally that g^(k)_P divides g^(k)_{P u λ} for any multiple k-rectangles P and any k-bounded partition λ. We give the factorization formula of such g^(k)_P and explicit formulas for g^(k)_{P u λ} / g^(k)_P in some cases.