Question 1: Find the prime factors of 48 and 60.

Answer: 48: 2^4 × 3; 60: 2^2 × 3 × 5

Question 2: Using prime factorization, determine the LCM of 36 and 45.

Answer: 2^2 × 3^2 × 5 = 180

Question 3: Calculate the LCM of 8, 12, and 20 using the prime factor method.

Answer: 2^3 × 3 × 5 = 120

Question 4: Explain the steps to find the LCM of 14 and 35 using prime factors.

Answer: Prime factors of 14: 2 × 7; of 35: 5 × 7. For LCM, take highest powers: 2 × 5 × 7 = 70.

Question 5: Find the LCM of 24 and 36. Show all prime factors and the final calculation.

Answer: 2^3 × 3^2 = 72

Question 6: A factory produces boxes of chocolates in packs of 18 and 24. What is the smallest number of chocolates that can be packed so that both types fit perfectly? Use prime factorization to find the answer.

Answer: Prime factors: 18 = 2 × 3^2; 24 = 2^3 × 3. LCM: 2^3 × 3^2 = 72 chocolates.

Question 7: Construct a triangle on the grid with sides of length 15 and 20, and find a third side that makes the triangle's perimeter divisible by the LCM of 15 and 20.

Answer: LCM of 15 and 20: 60. Perimeter should be multiple of 60, e.g., 15 + 20 + 25 = 60.

Question 8: Determine the LCM of 50 and 75 and explain why your answer is correct.

Answer: 2 × 5^2 × 3 = 150; because prime factors are 2, 3, 5^2, taking highest powers.

Question 9: Find the LCM of 12, 15, and 20. Show your prime factorization process.

Answer: 2^2 × 3 × 5 = 60

Question 10: Identify a common mistake made when calculating LCM via prime factors and correct it in the following example: 12 and 18 with prime factors 2^2 × 3 and 2 × 3^2.

Answer: Mistake: Using the lowest powers instead of the highest. Corrected: LCM is 2^2 × 3^2 = 36.

Question 11: Challenge: Find the LCM of 48, 60, and 72 using prime factorization. Which prime factors contribute to this LCM?

Answer: 2^4 × 3^2 × 5 = 720