Final answer:
To find the LCM of two numbers, you can list multiples or use prime factorization. Listing multiples involves finding common multiples, while prime factorization involves finding common prime factors. Examples are provided for both methods.
Step-by-step explanation:
To find the LCM of two numbers by listing multiples, you can list the multiples of each number until you find a common multiple. Let's solve a few examples:
- For 3 and 4, the multiples are: 3, 6, 9, 12... and 4, 8, 12, 16... The LCM is 12.
- For 8 and 9, the multiples are: 8, 16, 24... and 9, 18, 27... The LCM is 72.
- For 15 and 20, the multiples are: 15, 30, 45, 60... and 20, 40, 60... The LCM is 60.
- For 5 and 14, the multiples are: 5, 10, 15, 20... and 14, 28, 42, 56... The LCM is 70.
To find the LCM of two numbers using prime factorization, you need to factorize each number into its prime factors and then multiply the highest power of each common prime factor. Let's solve a few examples:
- For 30 and 10, the prime factors are: 30 = 2 × 3 × 5 and 10 = 2 × 5. The highest power of 2 is 1, the highest power of 3 is 1, and the highest power of 5 is 1. Therefore, the LCM is 2 × 3 × 5 = 30.
- For 42 and 126, the prime factors are: 42 = 2 × 3 × 7 and 126 = 2 × 3 × 3 × 7. The highest power of 2 is 1, the highest power of 3 is 2, and the highest power of 7 is 1. Therefore, the LCM is 2 × 3² × 7 = 126.
- For 12 and 9, the prime factors are: 12 = 2 × 2 × 3 and 9 = 3 × 3. The highest power of 2 is 2, and the highest power of 3 is 2. Therefore, the LCM is 2² × 3² = 36.
- For 8, 9, and 24, the prime factors are: 8 = 2 × 2 × 2, 9 = 3 × 3, and 24 = 2 × 2 × 2 × 3. The highest power of 2 is 3, and the highest power of 3 is 2. Therefore, the LCM is 2³ × 3² = 72.