**Core Concept**
Ulcerative colitis (UC) associated with primary sclerosing cholangitis (PSC) is a complex condition involving both gastrointestinal and hepatobiliary systems. The pathophysiology involves chronic inflammation and immune-mediated damage to the colonic mucosa and bile ducts.

**Why the Correct Answer is Right**
Patients with UC and PSC have an increased risk of developing colorectal cancer (CRC), which is a well-established association. This is thought to be due to chronic inflammation and genetic predisposition. Additionally, patients with PSC have an increased risk of developing cholangiocarcinoma, a type of bile duct cancer. The combination of UC and PSC also increases the risk of needing a colectomy, which is a surgical procedure to remove the colon.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because patients with UC and PSC do not have an increased risk of developing gastric cancer. In fact, patients with UC are at a slightly decreased risk of developing gastric cancer.
**Option B:** This option is incorrect because while patients with UC and PSC may have an increased risk of developing extraintestinal manifestations such as arthritis and skin lesions, this is not a unique association.
**Option C:** This option is incorrect because patients with UC and PSC do not have an increased risk of developing Crohn's disease, which is a different type of inflammatory bowel disease.

**Clinical Pearl / High-Yield Fact**
It is essential to screen patients with UC and PSC for colorectal cancer and cholangiocarcinoma through regular surveillance endoscopies and imaging studies.

**Correct Answer: A. Gastric cancer**