Question 1 What is the least common multiple (LCM) of 8, 10, and 12? Answer 1 To find the LCM of 8, 10, and 12: Write the prime factorization of each number: 8 = 2 3 8=2 3 10 = 2 × 5 10=2×5 12 = 2 2 × 3 12=2 2 ×3 Take the highest power of each prime factor: 2 3 2 3 (from 8) 3 3 (from 12) 5 5 (from 10) Multiply these together: 𝐿 𝐶 𝑀 = 2 3 × 3 × 5 = 8 × 3 × 5 = 120 LCM=2 3 ×3×5=8×3×5=120 Result: 120 Question 2 What is the Greatest Common Factor (GCF) of 4 and 12? Answer 1 Find the factors of each number: Factors of 4: 1, 2, 4 Factors of 12: 1, 2, 3, 4, 6, 12 The common factors are 1, 2, and 4. The greatest common factor is 4. Alternate Method: Prime factorization of 4: 2 2 2 2 Prime factorization of 12: 2 2 × 3 2 2 ×3 The highest common power of 2 is 2 2 2 2 , so GCF = 4. Question 3 How many composite numbers are there between 1 and 50? Answer 1 Composite numbers are numbers with more than two factors. The prime numbers between 1 and 50 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 (total 15 primes). Total numbers from 1 to 50: 50 Subtract primes and 1 (which is not composite): 50 − 15 − 1 = 34 50−15−1=34 Result: 34 composite numbers between 1 and 50 Question 4 How many composite numbers are there between 1 and 100? Answer 1 1 is not composite. There are 25 prime numbers between 1 and 100. Total numbers from 1 to 100: 100 Subtract primes and 1: 100 − 25 − 1 = 74 100−25−1=74 Result: 74 composite numbers between 1 and 100 Question 5 What is the least common multiple (LCM) of 4 and 7? Answer 1 To find the LCM of 4 and 7: List the multiples: Multiples of 4: 4, 8, 12, 16, 20, 24, 28, ... Multiples of 7: 7, 14, 21, 28, ... The smallest common multiple is 28.