Are $A,B,C$ coprime integers if $B=C$?

Question

Let $A,B,C$ be three coprime positive integers, i.e., there exist six integers $λ_{1},λ_{2},λ_{3},λ_{4},λ_{5},λ_{6}$ such that

$$λ_{1}A+λ_{2}B=1$$

$$λ_{3}A+λ_{4}C=1$$

$$λ_{5}B+λ_{6}C=1$$

I am asking what happens if $B=C$: are $A,B,C$ still three coprime positive integers or not?

Answer 1

It depends.

There are two conventions you can take. Let $S := \{a_1,\cdots,a_n\} \subseteq \Bbb Z$ be a set of integers. If you want to say the elements of $S$ are coprime, this can be interpreted two different ways:

• As a whole, they are coprime. This would mean that $\gcd(a_1,\cdots,a_n)=1$. That is, there are $\lambda_i \in \Bbb Z$ such that

$$\sum_{i=1}^n \lambda_i a_i = 1$$

• Alternately, they are pairwise coprime. That is, $\gcd(a_i,a_j) = 1$ for all $i\ne j$. Or, put differently, there are integers $\lambda_{i,j},\mu_{i,j} \in \Bbb Z$ such that $$\lambda_{i,j} a_i + \mu_{i,j} a_j = 1$$ for every $i \ne j$.

Suppose, then $S = \{A,B,C\}$. If $B=C$, and we don't "reduce" the set to the equivalent $S=\{A,B\}$, then $S$ is not coprime in the second sense (as $\gcd(B,B) = B$) unless $B=1$. However, it is in the first sense because

$$\gcd(A,B,B) = \gcd \big( \gcd(A,B)\; , \; B \;\big) = \gcd(1,B) = 1$$