Compute HCF and LCM of multiple integers by extracting minimal and maximal powers of prime factors.
Define HCF and LCM in terms of prime powers.
When calculating the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of two or more numbers, prime factorization provides a systematic method. This technique relies on identifying the underlying prime building blocks of each integer.
Visual comparison of smallest common vs greatest involved.
HCF extracts only the overlapping minimums, while LCM absorbs the total maximums.
Example 2: Find LCM and HCF of 6 and 20.
Find the LCM and HCF of 6 and 20 by the prime factorisation method.
Example 4: 6, 72, and 120.
Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.
Exercise 1.1 Q3(i) 12, 15, and 21.
Let us find the LCM and HCF of 12, 15, and 21 by applying the prime factorisation method. First, we express each number as a product of its prime factors: 12 = 2² × 3, 15 = 3 × 5, and 21 = 3 × 7.
To find the HCF, we identify the smallest power of each common prime factor across all three numbers. The only common prime factor is 3, so the HCF is .
To find the LCM, we multiply the greatest power of every prime factor involved in the numbers. Taking the highest powers of 2, 3, 5, and 7, we calculate 2² × 3 × 5 × 7.
Therefore, the LCM of these three integers is .
Test if student uses greatest instead of smallest power.
Two positive integers and are written as and , where and are prime numbers. What is the ?