1. **Problem statement:** Two trains leave the station at 9 am. One blows its whistle every 12 minutes, the other every 9 minutes. After the initial whistle at 9 am, we want to find when they will blow their whistles again at the exact same time. 2. **Formula and concept:** To find when two repeating events coincide, we find the Least Common Multiple (LCM) of their intervals. 3. **Prime factorization:** - 12 = 2 \times 2 \times 3 - 9 = 3 \times 3 4. **Calculate LCM:** The LCM is the product of the highest powers of all prime factors: $$\text{LCM} = 2^2 \times 3^2 = 4 \times 9 = 36$$ 5. **Interpretation:** The trains will blow their whistles together every 36 minutes after 9 am. 6. **Final answer:** $$9:00\ \text{am} + 36\ \text{minutes} = 9:36\ \text{am}$$ So, the trains will blow their whistles again at the exact same time at 9:36 am.