1. **Problem statement:** GreenTech has rectangular solar panels measuring 180 cm by 120 cm and wants to cut them into equal square segments without wastage. 2. **Mathematical technique:** To find the largest square size that can fit exactly into both dimensions, we need to find the Greatest Common Divisor (GCD) of 180 and 120. 3. **Step-by-step calculation of GCD:** - List factors or use Euclidean algorithm: $$\gcd(180,120) = \gcd(120, 180 \mod 120) = \gcd(120, 60)$$ $$\gcd(120, 60) = \gcd(60, 120 \mod 60) = \gcd(60, 0) = 60$$ 4. **Interpretation:** The largest square size is 60 cm. 5. **Explanation:** The GCD gives the largest number that divides both lengths exactly, ensuring no wastage when cutting squares. 6. **Problem statement:** Three energy-monitoring systems send reports every 6, 10, and 15 minutes. Find the earliest time all send reports simultaneously. 7. **Mathematical technique:** Find the Least Common Multiple (LCM) of 6, 10, and 15. 8. **Step-by-step calculation of LCM:** - Prime factorization: $$6 = 2 \times 3$$ $$10 = 2 \times 5$$ $$15 = 3 \times 5$$ - LCM is product of highest powers of all primes: $$\text{LCM} = 2 \times 3 \times 5 = 30$$ 9. **Interpretation:** The earliest time all systems send reports simultaneously is 30 minutes. 10. **Explanation:** LCM finds the smallest time interval divisible by all given times, so all events coincide.