1. The problem is to understand the concepts of GCD (Greatest Common Divisor) and LCM (Least Common Multiple) of two numbers. 2. The GCD of two integers is the largest positive integer that divides both numbers without leaving a remainder. 3. The LCM of two integers is the smallest positive integer that is divisible by both numbers. 4. Important formula relating GCD and LCM: $$\text{GCD}(a,b) \times \text{LCM}(a,b) = |a \times b|$$ 5. To find GCD, use the Euclidean algorithm: repeatedly replace the larger number by the remainder of dividing the larger by the smaller until the remainder is zero. The last non-zero remainder is the GCD. 6. To find LCM, use the formula: $$\text{LCM}(a,b) = \frac{|a \times b|}{\text{GCD}(a,b)}$$ 7. Example: Find GCD and LCM of 12 and 18. 8. Calculate GCD: - 18 ÷ 12 = 1 remainder 6 - 12 ÷ 6 = 2 remainder 0 - So, GCD(12,18) = 6 9. Calculate LCM: $$\text{LCM}(12,18) = \frac{12 \times 18}{6} = \frac{216}{6} = 36$$ 10. Final answers: - GCD(12,18) = 6 - LCM(12,18) = 36 This method works for any pair of integers.