1. **State the problem:** Divide and reduce to lowest terms the mixed numbers: $$1 \frac{1}{3} \div 2 \frac{5}{6}$$ 2. **Convert mixed numbers to improper fractions:** $$1 \frac{1}{3} = \frac{3 \times 1 + 1}{3} = \frac{4}{3}$$ $$2 \frac{5}{6} = \frac{6 \times 2 + 5}{6} = \frac{17}{6}$$ 3. **Division of fractions rule:** To divide by a fraction, multiply by its reciprocal: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$ 4. **Apply the rule:** $$\frac{4}{3} \div \frac{17}{6} = \frac{4}{3} \times \frac{6}{17}$$ 5. **Multiply numerators and denominators:** $$= \frac{4 \times 6}{3 \times 17} = \frac{24}{51}$$ 6. **Simplify the fraction:** Find the greatest common divisor (GCD) of 24 and 51, which is 3. $$\frac{24}{51} = \frac{\cancel{3} \times 8}{\cancel{3} \times 17} = \frac{8}{17}$$ 7. **Final answer:** $$1 \frac{1}{3} \div 2 \frac{5}{6} = \frac{8}{17}$$