Understanding Least Common Multiple (LCM)

To find the least common multiple (LCM) of two numbers, we can follow different methods, including prime factorization, listing multiples, or using the greatest common divisor (GCD). In this case, we will find the LCM of 150 and 500 using the GCD approach for efficiency.

Step 1: Prime Factorization

Let’s start by breaking down each number into its prime factors:

• 150 can be factored as follows:
  150 = 2 x 3 x 5²
• 500 can be factored as follows:
  500 = 2² x 5³

Step 2: Apply the LCM Formula

The formula to find the LCM using prime factors is:

LCM(a, b) = (p₁^m₁ × p₂^m₂ × … × p_n^m_n)

Where p represents the prime factors and m represents the highest power of each prime number appearing in the factorization of a and b.

From our prime factorizations:

• For 2: the highest power is 2²
• For 3: the highest power is 3¹
• For 5: the highest power is 5³

Step 3: Calculate the LCM

Now let’s calculate the LCM:

LCM(150, 500) = 2² × 3¹ × 5³

Calculating this gives:

• 2² = 4
• 3¹ = 3
• 5³ = 125

Multiplying these together:

4 × 3 = 12

12 × 125 = 1500

Conclusion

Therefore, the least common multiple (LCM) of 150 and 500 is 1500.