Step-by-step solution for dividing algebraic fractions, demonstrating factoring and reciprocal multiplication.

Problem:

We are tasked with solving the division of algebraic fractions:
\[
\frac{x - 3}{15} \div \frac{x^2 - 3x}{5}
\]

Step-by-Step Solution:

#### Step 1: Understand the division of fractions
The general rule for dividing fractions is:
\[
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}
\]
This means we take the reciprocal of the second fraction and then multiply.

#### Step 2: Rewrite the problem using the reciprocal
Given:
\[
\frac{x - 3}{15} \div \frac{x^2 - 3x}{5}
\]
We rewrite it as:
\[
\frac{x - 3}{15} \times \frac{5}{x^2 - 3x}
\]

#### Step 3: Factorize where possible
- The numerator \( x^2 - 3x \) can be factored:
\[
x^2 - 3x = x(x - 3)
\]
So, the expression becomes:
\[
\frac{x - 3}{15} \times \frac{5}{x(x - 3)}
\]

#### Step 4: Simplify the expression
Now, we have:
\[
\frac{x - 3}{15} \times \frac{5}{x(x - 3)}
\]
We can cancel out the common factor \( x - 3 \) in the numerator and denominator:
\[
\frac{\cancel{x - 3}}{15} \times \frac{5}{x \cdot \cancel{x - 3}} = \frac{1}{15} \times \frac{5}{x}
\]

#### Step 5: Multiply the remaining fractions
Multiply the numerators together and the denominators together:
\[
\frac{1}{15} \times \frac{5}{x} = \frac{1 \cdot 5}{15 \cdot x} = \frac{5}{15x}
\]

#### Step 6: Simplify the fraction
Simplify \( \frac{5}{15x} \) by dividing the numerator and the denominator by their greatest common divisor, which is 5:
\[
\frac{5}{15x} = \frac{1}{3x}
\]

Final Answer:

\[
\boxed{\frac{1}{3x}}
\]