1. **State the problem:** Find the greatest common factor (GCF) of 441693 and 819. 2. **Recall the formula and method:** The GCF of two numbers is the largest positive integer that divides both numbers without leaving a remainder. We use the Euclidean algorithm, which involves repeated division and taking remainders. 3. **Apply the Euclidean algorithm:** $$441693 \div 819 = 539 \text{ remainder } 462$$ 4. Now find GCF(819, 462): $$819 \div 462 = 1 \text{ remainder } 357$$ 5. Next find GCF(462, 357): $$462 \div 357 = 1 \text{ remainder } 105$$ 6. Next find GCF(357, 105): $$357 \div 105 = 3 \text{ remainder } 42$$ 7. Next find GCF(105, 42): $$105 \div 42 = 2 \text{ remainder } 21$$ 8. Next find GCF(42, 21): $$42 \div 21 = 2 \text{ remainder } 0$$ 9. When the remainder is 0, the divisor at this step is the GCF. So, the GCF is 21. **Final answer:** $$\boxed{21}$$