To find the least common multiple (LCM) of two numbers, we may use different methods. One such method is the prime factorization method where you prime factorize each number and take the highest powers of all the prime numbers involved to find the LCM. Let me guide you through the process step by step:
Step 1: Prime Factorize Each Number
First, we need to break down each number into its prime factors.
For 120:
120 = 2 × 2 × 2 × 3 × 5
120 = 2^3 × 3 × 5
For 150:
150 = 2 × 3 × 5 × 5
150 = 2 × 3 × 5^2
Step 2: Identify All Prime Factors
Now, we list out all the prime factors from both decompositions and identify the highest powers of these primes.
In this case, the prime factors are 2, 3, and 5.
- For the number 2, the highest power from both numbers is 2^3 (from 120).
- For the number 3, the highest power from both numbers is 3 (both have the same power of 3).
- For the number 5, the highest power from both numbers is 5^2 (from 150).
Step 3: Multiply the Highest Powers of All Primes
We now multiply these highest powers toget the LCM of 120 and 150.
LCM = 2^3 × 3 × 5^2
LCM = 8 × 3 × 25
LCM = 24 × 25
LCM = 600
So, the least common multiple of 120 and 150 is 600.
