Finding the least common multiple (LCM) of two numbers is a straightforward process, and it can be done using a couple of different methods. Here, we’ll explain the step-by-step process using the prime factorization method and the relationship between the LCM and the greatest common divisor (GCD).
Method 1: Prime Factorization
- Factor Each Number: Begin by finding the prime factorization of each number. For instance, if you’re finding the LCM of 12 and 18, the prime factorization of 12 is 2² × 3¹, and the prime factorization of 18 is 2¹ × 3².
- List All Prime Factors: Write down all unique prime factors from both numbers. For 12 and 18, the unique prime factors are 2 and 3.
- Choose the Highest Power: For each prime factor, take the highest exponent that appears in either factorization. For 2, the highest power is 2² (from 12). For 3, it’s 3² (from 18).
- Multiply Them Together: Multiply these highest powers together to get the LCM. In this case, LCM = 2² × 3² = 4 × 9 = 36.
Method 2: Using GCD
You can also find the LCM using the relationship between LCM and GCD:
LCM(a, b) = (a × b) / GCD(a, b)
- Find the GCD: First, find the greatest common divisor (GCD) of the two numbers. For 12 and 18, the GCD is 6.
- Calculate the LCM: Substitute the values into the LCM formula. LCM = (12 × 18) / 6 = 216 / 6 = 36.
Regardless of the method you choose, the least common multiple of 12 and 18 is 36. This number is the smallest multiple that can be evenly divided by both original numbers.