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Using a plastic putty knife, begin scraping and pulling out the old caulk that was just cut. The plastic putty knife is effective and won't scratch the surfaces. Work your way along the entire joint, digging out as much of the old caulk as possible. You may need to alternate between the putty knife and the utility knife to remove all the pieces. Place a rag over the drain to catch the debris. The goal is to create a clean, open gap for the new caulk to adhere to.