18 and 27 LCM =
The Correct Answer and Explanation is:
The Least Common Multiple (LCM) of 18 and 27 is 54.
Explanation:
To find the LCM of two numbers, we look for the smallest number that is a multiple of both. There are several ways to find the LCM, including listing multiples or using prime factorization. Here, we’ll use prime factorization and also verify using multiples.
Step 1: Prime Factorization
- 18 = 2 × 3 × 3 = 2 × 3²
- 27 = 3 × 3 × 3 = 3³
To find the LCM, take the highest powers of all the prime factors involved.
- From 18 and 27, the prime factors are 2 and 3.
- The highest power of 2 is 2¹ (from 18).
- The highest power of 3 is 3³ (from 27).
So the LCM is:
LCM = 2¹ × 3³ = 2 × 27 = 54
Step 2: Listing Multiples (for verification)
- Multiples of 18: 18, 36, 54, 72, 90…
- Multiples of 27: 27, 54, 81, 108…
The smallest common multiple is 54, which confirms our answer.
Why this works
The LCM represents the smallest number that both original numbers divide evenly into. This is useful in problems involving synchronization of cycles, such as finding when two traffic lights with different blinking intervals will blink at the same time.
By taking the highest powers of all prime factors, we ensure that both original numbers divide the LCM exactly. Using only the lowest powers would not capture the full divisibility requirement for both numbers.
In summary, the LCM of 18 and 27 is 54. This was found by taking the highest powers of their prime factors. Understanding the LCM is important in arithmetic, algebra, and real-life situations involving common intervals or repeated patterns.
