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How can prime factorization be used to find a least common multiple?

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To find the least common multiple (LCM) of a set of integers using prime factorization, you first express each number as a product of its prime numbers raised to their respective powers.1

Once each number is factored, the LCM is calculated by identifying every distinct prime factor that appears in any of the numbers. For each of these primes, you select the highest exponent (the largest power) with which it occurs across all the factorizations. Multiplying these prime powers together yields the LCM.1

For example, given two numbers:
* $a_{1} = 3, 000 = 2^{3} \times 3^{1} \times 5^{3}$
* $a_{2} = 2, 646 = 2^{1} \times 3^{3} \times 7^{2}$

The LCM is found by taking the highest power of each prime present: $2^{3}$ , $3^{3}$ , $5^{3}$ , and $7^{2}$ . Multiplying these results in $2^{3} \times 3^{3} \times 5^{3} \times 7^{2} = 1, 323, 000$ .1

This method relies on the fundamental theorem of arithmetic, which ensures that every composite number has a unique prime factorization.1