$\gcd$ and $\text{lcm}$ of more than $2$ positive integers

Question

For any two positive integers ${n_1,n_2}$, the relationship between their greatest common divisor and their least common multiple is given by

$$\text{lcm}(n_1,n_2)=\frac{n_1 n_2}{\gcd(n_1,n_2)}$$

If I have a set of $r$ positive integers ${n_1,n_2,n_3,...,n_r}$, does the same relationship hold? Is it true that

$$\text{lcm}(n_1,n_2,n_3,...,n_r)=\frac{\prod_{i=1}^r n_i}{gcd(n_1,n2,n_3,...,n_r)}$$

I feel like this should be easy to prove, but I'm struggling to get a handle on it.

Observe that $\text {gcd} (a,b,c) = \text {gcd} (\text {gcd} (a,b),c).$ — little o, Mar 15 '19 at 13:15

score 1 · Answer 1 · answered Mar 15 '19 at 13:16

1

Counterexample: $n_1=2,\,n_2=4,\,n_3=8$.

answered Mar 15 '19 at 13:16

J.G.

1 Answers1