1. We are given the orbital periods of Saturn, Jupiter, and Uranus as 12, 30, and 84 years respectively. 2. The problem asks when all three planets will align again to the same positions as today. 3. This is a problem of finding the least common multiple (LCM) of their orbital periods. 4. First, find the prime factorizations: - 12 = $2^2 \times 3$ - 30 = $2 \times 3 \times 5$ - 84 = $2^2 \times 3 \times 7$ 5. The LCM is found by taking the highest power of each prime factor: - Highest power of 2 is $2^2$ - Highest power of 3 is $3$ - Include 5 once (from 30) - Include 7 once (from 84) 6. So, the LCM is: $$\text{LCM} = 2^2 \times 3 \times 5 \times 7 = 4 \times 3 \times 5 \times 7$$ 7. Calculate: $$4 \times 3 = 12$$ $$12 \times 5 = 60$$ $$60 \times 7 = 420$$ 8. Therefore, all three planets will align again in 420 years from now. **Final answer:** 420 years