So no such angle exists that is multiple of 45 and a multiple of 18, except multiples of 90, which are invalid. - Tacotoon

Understanding the Context

An angle that is a multiple of both 45 and 18 must be a common multiple of these two numbers. To find such angles, we calculate the least common multiple (LCM) of 45 and 18, which serves as the smallest angle satisfying both conditions.

• Prime factorization:
  
  • 45 = 3² × 5
    
  • 18 = 2 × 3²
    
• LCM(45, 18) = 2 × 3² × 5 = 90

Thus, the smallest angle satisfying both conditions is 90°, and all valid angles are integer multiples of 90°:
90°, 180°, 270°, 360°, ...

Why Multiples of 45 and 18 Would Otherwise Be Invalid

Key Insights

The requirement for an angle to be a multiple of both 45° and 18° ensures that it supports symmetry or division compatible with both rotational divisions. However, if the number is not a multiple of 90°, it fails to perfectly align with the least common base rhythm established by LCM.

• For example:
  
  • 45° × 2 = 90° → valid factor of 90 (valid)
    
  • 18 × 5 = 90° → also valid
    
  • 45 × 2 = 90°, 18 × 2 = 36° → not common
    
  • 45° × 4 = 180° → valid multiple
    
  • 18° × 5 = 90° → coincidentally valid, but 180°, 270°, etc., follow the same logic

Thus, while some arbitrary products may coincidentally equal 90°, only multiples of 90° consistently fulfill both divisibility conditions without falling into the invalid zone of multiples of 90°—a boundary that represents misalignment in harmonious rotational symmetry.

Excluding Multiples of 90°: A Deliberate Filter

Although 180°, 270°, 360°, etc., are valid angles under both 45° and 18° constraints, the problem explicitly excludes them. This exclusion likely stems from a practical or conceptual boundary—perhaps emphasizing angles that maintain distinct angular “modes” or prevent degenerate cases (e.g., overlapping symmetry). In many geometric or engineering contexts, multiples of 90° are suppressed from consideration when specific angular rhythms are targeted.

Final Thoughts

Alternatively, excluding multiples of 90° may streamline problem-solving by focusing only on the least fundamental base unit (90°), reinforcing clarity and consistency in analysis.

Practical Takeaways

• The LCM(45, 18) = 90 sets 90° as the foundational angle satisfying both divisibility conditions.
  
• Multiples of 90° (e.g., 180°, 270°, 360°) are valid and fully compliant.
  
• Pure multiples of 45 and 18 that are not multiples of 90° fail to meet the dual-multiple criteria simultaneously.
  
• When both 45 and 18-degree divisions align, only harmonious angles like 90° work—making 90° the exclusive breakpoint.

Conclusion

While mathematical multiples of 45 and 18 produce several common angles like 90°, 180°, and others, only multiples of 90° consistently fulfill both conditions simultaneously without ambiguity. Avoiding these exceptions preserves rotational integrity and avoids dimensional conflicts. So yes, no angle other than the multiples of 90°—excluding those invalid crosses—truly satisfies the dual multiple requirement. Embracing 90° provides clarity, precision, and mathematical purity in angular analysis.