Hazak u'barukh is a phrase commonly said after one does something in the synagogue (e.g., after an aliyah). What is its origin?
The Rema 139:11 says To say Chazak from the passuk in Yehoshua that says Chazak vametz .The passuk before it says that Torah should not leave your mouth and it will be a blessing for you. So there are those who say Chazak u'baruch and others answer Chazak vametz.
The Kaf Hachaim 139:56 brings down the minhag to say Chazak U'baruch from this Rema.