Find smallest θ such that θ = 18n, θ = 45k, and θ mod 90 ≠ 0.
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- n = 20 → 360° → 360 ÷ 90 = 4 → invalid.
- But 45° is target, and no integer n gives 18n = 45 → so 45° is unreachable in 18° steps?
- Wait — the arm rotates incrementally, so position must be exact. But 45 and 18 have LCM:
- So 18n = 45k → 2n = 5k → n = 5m, k = 2m → θ = 18×5m = 90m° → always multiple of 90!
- So every multiple of 18° that is also multiple of 45 must be multiple of LCM(18,45).
- LCM(18,45) = (18×45)/gcd(18,45) = 810 / 9 = 90.
