Master finding the HCF and LCM of numbers using prime factorization, and apply them to real-world grouping and timing problems.
To find the Highest Common Factor (HCF) of two numbers, write each as a product of prime factors, then multiply together only the factors common to both.
Find the HCF of 36 and 60 using prime factorization.
36 = 2×2×3×3, 60 = 2×2×3×5; common factors: 2×2×3 = 12
Find the HCF of 48 and 18 using prime factorization.
48 = 2×2×2×2×3, 18 = 2×3×3; common factors: 2×3 = 6
To find the LCM, take every prime factor that appears in either number, using the highest power of each. LCM problems often come up in situations like syncing repeating events.
Find the LCM of 12 and 18 using prime factorization.
12 = 2²×3, 18 = 2×3²; LCM = 2²×3² = 36
Two bells ring every 6 and 8 minutes. If they ring together now, when will they next ring together?
LCM of 6 and 8 = 24, so they'll ring together again in 24 minutes
1. Find the HCF of 24 and 36 using prime factorization.
2. Find the HCF of 45 and 75 using prime factorization.
3. Find the LCM of 8 and 12 using prime factorization.
4. Find the LCM of 15 and 20 using prime factorization.
5. A shopkeeper wants to pack 30 pens and 45 pencils into identical bundles with no items left over, using the maximum bundle size. How many bundles will there be?