1. **Problem statement:** We want to find how often Jupiter, Saturn, and Neptune will align in the same position in the night sky as seen from Earth. 2. **Given data:** - Jupiter's revolution period: $12$ years - Saturn's revolution period: $30$ years - Neptune's revolution period: $165$ years 3. **Concept:** The planets align when their positions repeat simultaneously, which happens at the least common multiple (LCM) of their revolution periods. 4. **Calculate the LCM:** - Prime factorization: - $12 = 2^2 \times 3$ - $30 = 2 \times 3 \times 5$ - $165 = 3 \times 5 \times 11$ - LCM takes the highest powers of all primes: - $2^2$ (from 12), $3$ (common), $5$ (from 30 and 165), $11$ (from 165) - So, $$\text{LCM} = 2^2 \times 3 \times 5 \times 11 = 4 \times 3 \times 5 \times 11$$ 5. **Calculate the product:** $$4 \times 3 = 12$$ $$12 \times 5 = 60$$ $$60 \times 11 = 660$$ 6. **Answer:** The three planets will align every $660$ years. **Final answer:** 660 years