1. **State the problem:** Simplify the expression $$\frac{a}{a-b} \div \frac{a}{a-c} \div \frac{a}{a-d}$$. 2. **Recall the rule:** Dividing by a fraction is the same as multiplying by its reciprocal. 3. **Rewrite the expression:** $$\frac{a}{a-b} \times \frac{a-c}{a} \times \frac{a-d}{a}$$ 4. **Multiply numerators and denominators:** $$\frac{a \times (a-c) \times (a-d)}{(a-b) \times a \times a}$$ 5. **Simplify by canceling common factors:** The numerator has one $a$, the denominator has $a \times a = a^2$. Cancel one $a$ from numerator and denominator: $$\frac{(a-c)(a-d)}{a(a-b)}$$ 6. **Final simplified form:** $$\frac{(a-c)(a-d)}{a(a-b)}$$ This is the simplified expression.