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To find the Least Common Multiple (LCM) of any set of numbers, you start with Prime Factorization. Prime factorization means breaking down each number into its prime factors, which are numbers that are only divisible by 1 and themselves.
For example, with our given numbers:
- For 8: Break it down into prime factors, which gives us: \[8 = 2 \times 2 \times 2 = 2^3.\]
- For 10: The prime factors are: \[10 = 2 \times 5.\]
- For 12: The prime factors are: \[12 = 2 \times 2 \times 3 = 2^2 \times 3.\]
Prime factorization helps us understand the building blocks of each number. We need these prime factors to find our LCM.

After prime factorization, the next step is to identify the highest powers of all the primes that appear in the factorizations. You need these highest powers because the LCM must include the largest version of each prime factor.
Here’s how it works for our example:
- From the number 8, the highest power of 2 is \[2^3.\]
- From the number 12, the highest power of 3 is \[3^1.\]
- From the number 10, the highest power of 5 is \[5^1.\]
Now that we have all the highest powers of the prime numbers, we are ready to use them in calculating the LCM.

Remember, the key point is to use the absolute highest power found among all numbers being considered.

Finally, we calculate the LCM by multiplying these highest powers together. This step combines all the work we've done so far.
Let’s see this in action with our example:
- Take the highest power of 2 (which is 2^3), the highest power of 3 (which is 3^1), and the highest power of 5 (which is 5^1).
Now multiply them together to find the LCM: \[LCM = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120.\]

So, the LCM of 8, 10, and 12 is 120. This number is the smallest multiple that all the original numbers divide evenly into. By following the steps of prime factorization, identifying highest powers, and calculating the LCM, you can easily find the least common multiple for any group of numbers.