Question 1 To find the least common multiple of a pair of numbers, 12 and 16, using the multiples method. Answer 1 Given pair of numbers: 12, 16. Concept Used: The Least Common Multiple (LCM) is the smallest multiple that is completely divisible by the given numbers without any remainder. To find the LCM using the multiples method: List out multiples of each number and find the smallest multiple that is common between them. Solution: Multiples of 12: 12, 24, 36, 48, 60, ... Multiples of 16: 16, 32, 48, 64, ... The smallest common multiple between 12 and 16 is 48. Answer: The LCM of 12 and 16 is 48. Question 2 The difference between the GCF and LCM of two numbers. Answer 1 The Greatest Common Factor (GCF) is the largest number that divides both numbers. For example, if the numbers are 10 and 20: Factors of 10: 1, 2, 5, 10 Factors of 20: 1, 2, 4, 5, 10, 20 The common factors are: 1, 2, 5, and 10, so GCF = 10. The Least Common Multiple (LCM) is the smallest number that both numbers can divide into. For example, if the numbers are 6 and 12: Multiples of 6: 6, 12, 18, 24, ... Multiples of 12: 12, 24, 36, ... The smallest common multiple is 12, so LCM = 12. Answer: The difference between the GCF and LCM of 6 and 12 is 2. Question 3 Find the least common multiple of the given numbers: 8 and 12. Answer 1 Using the prime factorization method: 8 = 2 × 2 × 2 = 2 3 2×2×2=2 3 12 = 2 × 2 × 3 = 2 2 × 3 2×2×3=2 2 ×3 To find the LCM, take the highest powers of each prime factor: 2 3 = 8 2 3 =8 3 1 = 3 3 1 =3 LCM = 8 × 3 = 24 8×3=24 Answer: The least common multiple of 8 and 12 is 24. Question 4 What is the prime factorization of 125? Answer 1 To find the prime factorization of 125: Divide by 5: 125 ÷ 5 = 25 125÷5=25 Divide 25 by 5: 25 ÷ 5 = 5 25÷5=5 Divide 5 by 5: 5 ÷ 5 = 1 5÷5=1 Prime Factorization: 125 = 5 × 5 × 5 = 5 3 125=5×5×5=5 3 Answer: The prime factorization of 125 is 5 3 5 3 . Question 5 Use a direct proof to show that the product of two odd numbers is odd. Answer 1 To prove this, let 𝑎 a and 𝑏 b be two odd numbers. An odd number can be expressed as 2 𝑛 + 1 2n+1 where 𝑛 n is an integer. So, let: 𝑎 = 2 𝑚 + 1 a=2m+1 𝑏 = 2 𝑘 + 1 b=2k+1, where 𝑚 m and 𝑘 k are integers. The product 𝑎 𝑏 ab is: 𝑎 𝑏 = ( 2 𝑚 + 1 ) ( 2 𝑘 + 1 ) ab=(2m+1)(2k+1) Expanding this, we get: 𝑎 𝑏 = 4 𝑚 𝑘 + 2 𝑚 + 2 𝑘 + 1 = 2 ( 2 𝑚 𝑘 + 𝑚 + 𝑘 ) + 1 ab=4mk+2m+2k+1=2(2mk+m+k)+1 Since 2 𝑚 𝑘 + 𝑚 + 𝑘 2mk+m+k is an integer, let’s call it 𝑝 p: 𝑎 𝑏 = 2 𝑝 + 1 ab=2p+1 Thus, 𝑎 𝑏 ab is in the form 2 𝑝 + 1 2p+1, which is an odd number. Conclusion: The product of two odd numbers is odd.