Proposition: Factorization of Greatest Common Divisor and Least Common Multiple

Let $n,m$ be natural numbers with $n\ge 1,$ $m\ge 1,$ and the factorizations $n=\prod_{i=1}^\infty p_i^{e_i}$ and $m=\prod_{i=1}^\infty p_i^{f_i}.$ Then the greatest common divisor $\gcd(n,m)$ and the least common multiple $\operatorname{lcm}(n,m)$ have the following factorizations:

$$\gcd(n,m)=\prod_{i=1}^\infty p_i^{\min(e_i,f_i)},\quad \operatorname{lcm}(n,m)=\prod_{i=1}^\infty p_i^{\max(e_i,f_i)},$$

where $\min(e_i,f_i)$ and $\max(e_i,f_i)$ are the minimum or the maximum of the respective exponents $e_i,f_i.$

Proofs: 1


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References

Bibliography

  1. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927