Final answer:
The priority problem for a patient with peripheral artery disease (PAD) and a foot ulcer is impaired skin integrity, which requires immediate attention to prevent serious infections and further complications.
Step-by-step explanation:
In a patient diagnosed with arterial occlusive disease, commonly known as peripheral artery disease (PAD), the priority client problem when admitted to the hospital with a foot ulcer is impaired skin integrity. A foot ulcer represents a breakdown in the skin which can lead to serious infections, such as gas gangrene, especially in individuals with conditions that impair circulation and wound healing, like diabetes. In the context of PAD, where blood flow is already compromised due to the narrowing of the arteries, maintaining skin integrity is critical to prevent further complications like infection or necrosis that might even lead to amputation. Other problems like activity intolerance, ineffective health maintenance, and risk for peripheral neuropathy are also important, but the immediate threat to life and limb presented by a foot ulcer makes impaired skin integrity the most urgent issue requiring attention.