Find the LCM of the following numbers by prime factorization method: 18 and 27.
The Correct Answer and Explanation is:
To find the least common multiple (LCM) of 18 and 27 using the prime factorization method, we first need to express each number as a product of prime factors.
Step 1: Prime Factorization of 18 and 27
Prime factorization of 18:
Start by dividing 18 by the smallest prime number (2).
18 ÷ 2 = 9
Now, factor 9:
9 ÷ 3 = 3
3 ÷ 3 = 1
So, the prime factorization of 18 is:
18 = 2 × 3²
Prime factorization of 27:
Start by dividing 27 by the smallest prime number (3).
27 ÷ 3 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1
So, the prime factorization of 27 is:
27 = 3³
Step 2: Determine the LCM
To find the LCM, we take the highest powers of all prime factors present in both numbers. We compare the prime factorizations of 18 and 27:
- The prime factors of 18 are 2 and 3².
- The prime factor of 27 is 3³.
Now, the LCM is obtained by taking the highest power of each prime factor:
- The highest power of 2 is 2¹ (since 2 only appears in 18).
- The highest power of 3 is 3³ (since 27 has 3³, which is higher than 3²).
Thus, the LCM is:
LCM = 2¹ × 3³ = 2 × 27 = 54
Step 3: Conclusion
The least common multiple of 18 and 27 is 54.
Explanation:
The prime factorization method involves breaking down each number into its prime factors and then taking the highest powers of each prime factor from both numbers. This ensures that the LCM is the smallest number that both 18 and 27 divide into without a remainder. In this case, 54 is the smallest such number.
