Question 1: Find the prime factorization of 48.

Answer: 2^4 x 3

Question 2: Find the prime factorization of 180.

Answer: 2^2 x 3^2 x 5

Question 3: Calculate the LCM of 48 and 180 using the Prime Factor Method.

Answer: 2^4 x 3^2 x 5

Question 4: Explain the steps involved in finding the LCM of 36 and 48 using prime factors.

Answer: Prime factorise both numbers, identify the highest powers of each prime, then multiply these highest powers to get the LCM.

Question 5: The numbers 60 and 84 are given. Find their LCM by prime factorization.

Answer: 2^2 x 3 x 5 x 7

Question 6: Construct a pair of numbers with prime factorizations 2^3 x 3 and 2^2 x 3^2. Find their LCM.

Answer: 2^3 x 3^2

Question 7: A problem involves finding the LCM of 24, 36, and 60. Find the LCM using the Prime Factor Method.

Answer: 2^3 x 3^2 x 5

Question 8: Identify and correct the mistake in the following solution: The student found the prime factors of 40 as 2^3 and 5, then concluded the LCM of 40 and 80 as 2^3 x 5. Is this correct? Why or why not?

Answer: The mistake is in assuming the highest powers are 2^3 and 5; the correct highest power for 80 is 2^4, so the LCM should include 2^4, not 2^3.

Question 9: Find the LCM of 14 and 35 using prime factors and explain your method.

Answer: Prime factorize: 14=2 x 7, 35=5 x 7; highest powers: 2, 5, 7; LCM=2 x 5 x 7=70.

Question 10: Challenge: Find the LCM of 81 and 108 using prime factorization. What are the key considerations in this process?

Answer: 81=3^4, 108=2^2 x 3^3; highest powers are 3^4 and 2^2; LCM=2^2 x 3^4=162.