To find the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM) of 396 and 1080 using prime factorization, we first need to break down both numbers into their prime factors.
Step 1: Prime Factorization
For 396:
1. Divide by 2 (the smallest prime number): 396 ÷ 2 = 198
2. Divide 198 by 2: 198 ÷ 2 = 99
3. Now, divide 99 by 3: 99 ÷ 3 = 33
4. Divide 33 by 3 again: 33 ÷ 3 = 11
5. Finally, 11 is a prime number.
So, the prime factorization of 396 is: 2² × 3² × 11.
For 1080:
1. Divide by 2: 1080 ÷ 2 = 540
2. Divide 540 by 2: 540 ÷ 2 = 270
3. Divide 270 by 2: 270 ÷ 2 = 135
4. Now, divide 135 by 3: 135 ÷ 3 = 45
5. Divide 45 by 3: 45 ÷ 3 = 15
6. Finally, divide 15 by 3: 15 ÷ 3 = 5
7. Here, 5 is a prime number.
So, the prime factorization of 1080 is: 2³ × 3³ × 5.
Step 2: Finding HCF
The HCF is found by taking the lowest power of all prime factors present in both factorizations:
- For 2, the lowest power is 22 (from 396).
- For 3, the lowest power is 32 (from 396).
- 11 and 5 are not common.
Thus, the HCF = 2² × 3² = 4 × 9 = 36.
Step 3: Finding LCM
The LCM is found by taking the highest power of all prime factors present in either factorization:
- For 2, the highest power is 23 (from 1080).
- For 3, the highest power is 33 (from 1080).
- For 5, the highest power is 51 (from 1080).
- For 11, the highest power is 111 (from 396).
Thus, the LCM = 2³ × 3³ × 5 × 11 = 8 × 27 × 5 × 11 = 1080.
Final Results
The HCF of 396 and 1080 is 36, and the LCM is 11880.