Finding HCF and LCM by Prime Factorization
To find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of the numbers 90 and 144 using the prime factorization method, follow these steps:
Step 1: Prime Factorization
First, we need to express both numbers as a product of their prime factors.
Prime factorization of 90:
- 90 can be divided by 2 (the smallest prime number): 90 ÷ 2 = 45.
- Next, 45 can be divided by 3: 45 ÷ 3 = 15.
- Then, 15 can also be divided by 3: 15 ÷ 3 = 5.
- Finally, 5 is a prime number and can’t be divided further.
Therefore, the prime factorization of 90 is:
90 = 21 × 32 × 51
Prime factorization of 144:
- 144 can be divided by 2: 144 ÷ 2 = 72.
- 72 can be divided by 2 again: 72 ÷ 2 = 36.
- 36 can be divided by 2: 36 ÷ 2 = 18.
- 18 can be divided by 2 once more: 18 ÷ 2 = 9.
- Finally, 9 can be divided by 3: 9 ÷ 3 = 3.
- 3 is a prime number, so we stop here.
The prime factorization of 144 is:
144 = 24 × 32
Step 2: Finding the HCF
The HCF (also known as GCD) is found by taking the lowest power of each common prime factor.
- Common prime factors: 2 and 3.
- For 2: the minimum power is 21.
- For 3: the minimum power is 32.
Thus, the HCF of 90 and 144 is:
HCF = 21 × 32 = 2 × 9 = 18
Step 3: Finding the LCM
The LCM is found by taking the highest power of each prime factor present in either number.
- For 2: the maximum power is 24.
- For 3: the maximum power is 32.
- For 5: since 5 only appears in the factorization of 90, we take 51.
Thus, the LCM of 90 and 144 is:
LCM = 24 × 32 × 51 = 16 × 9 × 5 = 720
Summary
In conclusion, using the prime factorization method:
- HCF of 90 and 144 is 18.
- LCM of 90 and 144 is 720.