1. **Problem 1: Find the Lowest Common Multiple (LCM) of 6 and 10.** 2. The LCM of two numbers is the smallest positive integer that is divisible by both numbers. 3. To find the LCM, we can use the prime factorization method: - Prime factors of 6: $2 \times 3$ - Prime factors of 10: $2 \times 5$ 4. Take the highest powers of all prime factors appearing in either number: - For 2: highest power is $2^1$ - For 3: highest power is $3^1$ - For 5: highest power is $5^1$ 5. Multiply these together: $$\text{LCM} = 2 \times 3 \times 5 = 30$$ --- 6. **Problem 2: Amelia's number is a factor of both 18 and 12 and is odd. Find the largest such number.** 7. First, find the common factors of 18 and 12 by finding their Highest Common Factor (HCF): - Prime factors of 18: $2 \times 3^2$ - Prime factors of 12: $2^2 \times 3$ 8. The HCF is the product of the lowest powers of common primes: - For 2: lowest power is $2^1$ - For 3: lowest power is $3^1$ $$\text{HCF} = 2 \times 3 = 6$$ 9. Factors of 6 are 1, 2, 3, 6. Among these, the odd factors are 1 and 3. 10. The largest odd factor is $3$. --- 11. **Problem 3: Find the Highest Common Factor (HCF) of 8 and 20.** 12. Prime factors: - 8: $2^3$ - 20: $2^2 \times 5$ 13. The HCF is the product of the lowest powers of common primes: - For 2: lowest power is $2^2$ 14. So, $$\text{HCF} = 2^2 = 4$$