find the HCF and LCM of 36 , 54 and 90 using prime factorisation method
The Correct Answer and Explanation is:
To find the HCF (Highest Common Factor) and LCM (Least Common Multiple) of 36, 54, and 90 using the prime factorisation method, follow these steps:
Step 1: Prime factorisation
36
= 2 × 2 × 3 × 3
= 2² × 3²
54
= 2 × 3 × 3 × 3
= 2 × 3³
90
= 2 × 3 × 3 × 5
= 2 × 3² × 5
Step 2: HCF (Highest Common Factor)
To find the HCF, take the lowest power of each common prime factor.
Common prime factors: 2 and 3
- 2 appears as: 2² (in 36), 2¹ (in 54), 2¹ (in 90) → take 2¹
- 3 appears as: 3² (in 36), 3³ (in 54), 3² (in 90) → take 3²
So,
HCF = 2¹ × 3² = 2 × 9 = 18
Step 3: LCM (Least Common Multiple)
To find the LCM, take the highest power of each prime factor that appears in any number.
Prime factors: 2, 3, and 5
- 2 → highest power is 2² (from 36)
- 3 → highest power is 3³ (from 54)
- 5 → appears only in 90 as 5¹
So,
LCM = 2² × 3³ × 5 = 4 × 27 × 5 = 540
Final Answers:
HCF = 18
LCM = 540
Explanation
The prime factorisation method is a systematic way to find both the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of a set of numbers. This method involves expressing each number as a product of its prime factors.
First, perform the prime factorisation of each number. For 36, the prime factors are 2² × 3². For 54, they are 2 × 3³. For 90, the prime factorisation is 2 × 3² × 5. Once all numbers are expressed in terms of their prime factors, we can proceed to find the HCF and LCM.
To find the HCF, we look for common prime factors in all three numbers and take the lowest exponent of each. In this case, the common prime factors are 2 and 3. The lowest power of 2 across all numbers is 1, and for 3, it is 2. Multiplying these gives the HCF as 18.
For the LCM, we consider all prime factors present in any of the numbers and take the highest power of each. Here, the highest power of 2 is 2, for 3 it is 3, and 5 appears only once in 90. Thus, multiplying 2² × 3³ × 5 gives us 540 as the LCM.
Using this method ensures accuracy because it is based on the fundamental principle that any composite number can be uniquely expressed as a product of primes. It is particularly useful for larger numbers, where listing out factors or multiples is not practical.
