1. The problem states that Saturn orbits the sun every 12 years, Jupiter every 30 years, and Uranus every 84 years. We want to find the number of years from now when all three planets will align in the same positions as today. 2. This is a problem of finding the least common multiple (LCM) of the orbital periods since the planets align together again after a time equal to the LCM of their orbital times. 3. The orbital periods are 12, 30, and 84 years. We need to find $$\text{LCM}(12, 30, 84)$$. 4. First, find the prime factorization of each: - $$12 = 2^2 \times 3$$ - $$30 = 2 \times 3 \times 5$$ - $$84 = 2^2 \times 3 \times 7$$ 5. To find the LCM, take the highest powers of all primes from the factorizations: - For 2: highest power is $$2^2$$ - For 3: highest power is $$3$$ - For 5: highest power is $$5$$ - For 7: highest power is $$7$$ 6. Thus, $$\text{LCM} = 2^2 \times 3 \times 5 \times 7 = 4 \times 3 \times 5 \times 7$$ 7. Calculate this step-by-step: - $$4 \times 3 = 12$$ - $$12 \times 5 = 60$$ - $$60 \times 7 = 420$$ 8. Therefore, all three planets will align again in $$\boxed{420}$$ years from now.