1. Problem statement: Compute $4\left(\frac{1}{15}\right)\cdot39\cdot\left(\frac{1}{53}\right). 2. Formula and important rules: To multiply fractions multiply numerators and denominators using the rule $$\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}$$ Remember you can cancel common factors before multiplying to keep numbers small. 3. Convert the product into a single fraction by multiplying numerators and denominators. $$\frac{4\cdot1\cdot39\cdot1}{15\cdot53}$$ 4. Factor to find common factors: $39=3\cdot13$ and $15=3\cdot5$. 5. Cancel the common factor 3 before multiplying to simplify; show the cancellation explicitly. $$\frac{4\cdot1\cdot(\cancel{3}\cdot13)\cdot1}{(\cancel{3}\cdot5)\cdot53}$$ 6. After canceling the 3, simplify the remaining product. $$\frac{4\cdot13}{5\cdot53}$$ 7. Multiply numerator and denominator to get the result and check for further simplification. $$\frac{52}{265}$$ 8. Verify simplest form: 52 and 265 have no common factor greater than 1, so the fraction is in lowest terms. 9. Final answer: $\frac{52}{265}$.