1. The problem is to simplify the expression $\frac{a}{a-b} \div \frac{a}{a-c} \div \frac{a}{a-d}$. 2. Recall that dividing by a fraction is the same as multiplying by its reciprocal. So rewrite the expression as $$\frac{a}{a-b} \times \frac{a-c}{a} \times \frac{a-d}{a}.$$ 3. Now multiply the numerators and denominators: $$\frac{a \times (a-c) \times (a-d)}{(a-b) \times a \times a}.$$ 4. Simplify by canceling common factors. The numerator has $a$, and the denominator has $a \times a = a^2$. Cancel one $a$ from numerator and denominator: $$\frac{(a-c)(a-d)}{(a-b) a}.$$ 5. The simplified expression is $$\frac{(a-c)(a-d)}{a(a-b)}.$$ This is the final simplified form of the given expression.