Proof Greatest Common Divisor

Question

Question: Let $m, n, k \in \mathbb{Z}$ with $\operatorname{gcd}(m, n) = 1 = \operatorname{gcd}(n, k)$

Determine if it is true that necessarily $\operatorname{gcd}(m, k) = 1$.

Answer: The statement is FALSE.

Proof : [By Contradiction]

Let $m=3, n=5, k=6$.

The $\operatorname{gcd}(3,5)=1=\operatorname{gcd}(5,6)$ , then the $\operatorname{gcd}(3,6)\neq1$ .

Hence, the statement is FALSE.

Answer 1

Your proof is correct. Another example would be $m=k=2$ and $n=1$.