An apparatus for preventing stroke by occluding blood flow through a right carotid artery and a left carotid artery of a patient, comprising: a first occluding catheter that carries a first occluding balloon that has an inflated configuration that is configured to occlude flow to either the right carotid artery or the left carotid artery; and a second occluding catheter that carries a second occluding balloon that has an inflated configuration that is configured to occlude flow to the other one of the right carotid artery or the left carotid artery that is not occluded by the first occluding balloon; wherein the first occluding catheter and the first occluding balloon are made of a highly compliant material and have a floating capability to reach desired positions in the circulatory system of the patient by being naturally propelled to a first target location for occlusion via forward arterial blood flow of the circulatory system; wherein the second occluding catheter and the second occluding balloon are made of a highly compliant material and have a floating capability to reach desired positions in the circulatory system of the patient by being naturally propelled to a second target location for occlusion via forward arterial blood flow of the circulatory system, and wherein the apparatus further comprises an insertion device that defines a first occluding catheter channel and a second occluding catheter channel; wherein the first occluding catheter is located in the first occluding catheter channel and is movable through the first occluding catheter channel such that the first occluding catheter is movable relative to the insertion device; and wherein the second occluding catheter is located in the second occluding catheter channel and is movable through the second occluding catheter channel such that the second occluding catheter is movable relative to the insertion device and is movable relative to the first occluding catheter. a first occluding catheter that carries a first occluding balloon that has an inflated configuration that is configured to occlude flow to either the right carotid artery or the left carotid artery; and a second occluding catheter that carries a second occluding balloon that has an inflated configuration that is configured to occlude flow to the other one of the right carotid artery or the left carotid artery that is not occluded by the first occluding balloon; wherein the first occluding catheter and the first occluding balloon are made of a highly compliant material and have a floating capability to reach desired positions in the circulatory system of the patient by being naturally propelled to a first target location for occlusion via forward arterial blood flow of the circulatory system; wherein the second occluding catheter and the second occluding balloon are made of a highly compliant material and have a floating capability to reach desired positions in the circulatory system of the patient by being naturally propelled to a second target location for occlusion via forward arterial blood flow of the circulatory system, and wherein the apparatus further comprises an insertion device that defines a first occluding catheter channel and a second occluding catheter channel; wherein the first occluding catheter is located in the first occluding catheter channel and is movable through the first occluding catheter channel such that the first occluding catheter is movable relative to the insertion device; and wherein the second occluding catheter is located in the second occluding catheter channel and is movable through the second occluding catheter channel such that the second occluding catheter is movable relative to the insertion device and is movable relative to the first occluding catheter.