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To find the least common multiple (LCM) of any two numbers, the first step is to break down each number into its prime factors. Prime factors are the building blocks of a number and are prime numbers themselves. For example, the prime factors of 10 are 2 and 5, while the prime factors of 15 are 3 and 5. These factors are special because they cannot be broken down further into smaller whole numbers. Think of prime factors as the basic units that, when multiplied together, form your original number. This step is crucial because it lays the groundwork for the entire LCM calculation.

Once you have listed all the prime factors of each number, the next step is to identify the highest power of each prime factor. For the numbers 10 and 15, we have the prime factors 2, 3, and 5. Here’s how to proceed: If any prime factor appears in both numbers, you only need to take one instance of that factor at its highest power. In our example:

• The prime factor 2 appears only in 10.
• The prime factor 3 appears only in 15.
• The prime factor 5 appears in both 10 and 15, but we only count it once.

After identifying the relevant powers, we select the highest power of each prime factor to use in our next step. This method ensures that we capture the full set of nuances in both numbers.

Now that we know the highest power of each prime factor, we can calculate the LCM. The LCM is found by multiplying these highest powers together. In our scenario with the numbers 10 and 15: We have prime factors 2, 3, and 5. Each of these prime factors is raised to the power of 1 since they only appear once each. So, we calculate the LCM by multiplying them: \[ 2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30 \] Therefore, the least common multiple of 10 and 15 is 30. The LCM represents the smallest number that both original numbers can divide into without leaving a remainder. This makes it incredibly useful for solving problems involving multiple numbers meeting at common points or intervals.