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- But 45 is not divisible by 18. So can the robotic arm reach 45°? Only if 18n = 45 → n = 2.5 → not integer → impossible.
- In fact, only angles divisible by gcd(18,45) = 9° are reachable? No — by the linear combination: 18n - 45k = 0 → all reachable angles are multiples of 9, but specifically, since 18 and 45 share factor 9, reachable angles are multiples of 9°, but more precisely, the set is {9k | k ∈ ℤ}, but constrained by step size.
- The general reachable angles are all integer multiples of 9°, but only when achievable by 18n.
- Specifically, 18n = 45k → 2n = 5k → n = 5m, k = 2m → 18×5m = 90m° → so only multiples of 90° are reachable that are also multiples of 45.
- Thus, the only positions that are multiples of 45 and reachable by 18° increments are multiples of 90° — which are also multiples of 90, hence invalid.
- So is there no solution? But that can't be.
Solution: We are given the expression:
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