1. **Problem statement:** A green light flashes every 12 minutes, and a red light flashes every 45 minutes. Both lights flash together at 9:00 AM. We need to find the next time when both lights will flash together. 2. **Formula and concept:** The two lights will flash together at intervals equal to the Least Common Multiple (LCM) of their flashing intervals. 3. **Calculate the LCM:** - Prime factorization of 12: $12 = 2^2 \times 3$ - Prime factorization of 45: $45 = 3^2 \times 5$ 4. **Find LCM by taking the highest powers of all prime factors:** $$\text{LCM} = 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180$$ 5. **Interpretation:** The lights flash together every 180 minutes. 6. **Convert 180 minutes to hours:** $$180 \text{ minutes} = \frac{180}{60} = 3 \text{ hours}$$ 7. **Find the next time after 9:00 AM:** $$9:00 \text{ AM} + 3 \text{ hours} = 12:00 \text{ PM}$$ **Final answer:** The two lights will next flash together at 12:00 PM.