Find the LCM and HCF of 72, 80, and 120 by applying prime factorization.
The correct answer and explanation is:
Correct Answer:
- HCF (Highest Common Factor) = 8
- LCM (Lowest Common Multiple) = 720
Explanation:
To find the HCF and LCM of 72, 80, and 120 using prime factorization, start by breaking each number into its prime factors:
1. Prime Factorization:
- 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²
- 80 = 2 × 2 × 2 × 2 × 5 = 2⁴ × 5
- 120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5
2. HCF (Greatest Common Factor):
To find the HCF, take the lowest powers of all common prime factors:
- The common prime factor among all three numbers is 2.
- The lowest power of 2 that appears in all is 2³.
There are no other common prime factors among all three numbers.
Therefore, HCF = 2³ = 8
3. LCM (Lowest Common Multiple):
To find the LCM, take the highest powers of all prime factors that appear in any of the numbers:
- Prime factor 2: highest power is 2⁴ (from 80)
- Prime factor 3: highest power is 3² (from 72)
- Prime factor 5: highest power is 5 (from 80 and 120)
Now multiply these together:
LCM = 2⁴ × 3² × 5 = 16 × 9 × 5 = 720
Conclusion:
Using prime factorization, the HCF of 72, 80, and 120 is 8, and the LCM is 720. Prime factorization makes it easier to see the structure of each number and determine both the highest common factor and the least common multiple by comparing prime powers.