To find the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of the numbers 396 and 1080 using prime factorization, follow these steps:
Step 1: Prime Factorization
First, we need to determine the prime factors of each number.
Prime Factorization of 396:
1. Start with the smallest prime number, which is 2:
- 396 ÷ 2 = 198
- 198 ÷ 2 = 99
2. Next, we divide 99 by the next smallest prime, which is 3:
- 99 ÷ 3 = 33
- 33 ÷ 3 = 11
3. Finally, since 11 is a prime number, we can stop here.
So, the prime factorization of 396 is:
396 = 22 × 32 × 11
Prime Factorization of 1080:
1. Start with 2 again:
- 1080 ÷ 2 = 540
- 540 ÷ 2 = 270
- 270 ÷ 2 = 135
2. Now, divide 135 by 3:
- 135 ÷ 3 = 45
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
3. Since 5 is a prime number, we stop here.
Thus, the prime factorization of 1080 is:
1080 = 23 × 33 × 5
Step 2: Finding the HCF
The HCF is found by taking the lowest powers of all prime factors present in both factorizations.
- For the prime number 2, the lowest power is 22.
- For the prime number 3, the lowest power is 32.
- The prime number 11 is not present in the factorization of 1080, so we exclude it.
- The prime number 5 is not present in the factorization of 396, so we exclude it.
Now, we multiply the selected factors:
HCF = 22 × 32 = 4 × 9 = 36
Step 3: Finding the LCM
The LCM is found by taking the highest powers of all prime factors present in either factorization.
- For the prime number 2, the highest power is 23.
- For the prime number 3, the highest power is 33.
- For the prime number 5, the highest power is 51.
- For the prime number 11, the highest power is 111.
Now, we multiply the selected factors:
LCM = 23 × 33 × 5 × 11 = 8 × 27 × 5 × 11
Calculating step-by-step:
- 8 × 27 = 216
- 216 × 5 = 1080
- 1080 × 11 = 11880
Thus, the LCM is:
LCM = 11880
Final Results
Therefore, we have:
- HCF of 396 and 1080 is 36
- LCM of 396 and 1080 is 11880