Find the HCF and LCM of 99 121 363 by using prime factorization.
The correct answer and explanation is:
Correct Answer:
HCF = 11
LCM = 39699
Explanation:
To find the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM) of 99, 121, and 363 using prime factorization, each number is broken down into its prime factors.
Step 1: Prime Factorization
- 99 = 3 × 3 × 11 = 3² × 11
- 121 = 11 × 11 = 11²
- 363 = 3 × 121 = 3 × 11 × 11 = 3 × 11²
Step 2: Find the HCF
The HCF is the product of the lowest powers of common prime factors.
- The only common prime factor among all three numbers is 11.
- The lowest power of 11 is 11¹.
So, HCF = 11
Step 3: Find the LCM
The LCM is the product of the highest powers of all prime factors present in any of the numbers.
- Prime factors involved: 3 and 11
- Highest power of 3 is 3²
- Highest power of 11 is 11²
So,
LCM = 3² × 11² = 9 × 121 = 1089
Now multiply 1089 by any missing factor from the third number:
Since 99 is 3² × 11, and we already used both in the LCM, and
121 is 11², which is already in, and
363 is 3 × 11², again, all are included.
So LCM is:
LCM = 3² × 11² = 9 × 121 = 1089
But we are missing one more multiplication. Actually, the correct LCM should be the smallest number divisible by all three numbers. Let’s multiply all distinct primes with their highest powers:
3² = 9
11² = 121
LCM = 9 × 121 = 1089
Check:
- 1089 ÷ 99 = 11
- 1089 ÷ 121 = 9
- 1089 ÷ 363 = 3
So, LCM = 1089
Oops — correction: The LCM of 99, 121, and 363 using the prime factorizations:
Let’s use all highest powers:
99 = 3² × 11
121 = 11²
363 = 3 × 11²
So LCM = 3² × 11² = 9 × 121 = 1089
Final Answers:
HCF = 11
LCM = 1089