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?

The only way that $B=C$ can be coprime to each other is if $B=C=1$. — Greg Martin, Dec 16 '20 at 08:31
Perhaps OP means what happens if $B=C$ and the three equations are true. The last equation would then imply $\lambda_5+\lambda_6=\pm1$ and $B,C=\mp1$. — anon, Dec 16 '20 at 08:34
@GregMartin: I am confused due to the answer below. It seems that there is 2 definitions. — Safwane, Dec 16 '20 at 09:31
It's analogous to this question Should I use sets or tuples when dealing with linear dependence?. If $`A,B,C'$ denotes a multiset (repetitions allowed) then they are not pairwise coprime when $B = C$. Otoh, if it denotes a set then they are pairwise coprime since ${A,B,B} = {A,B}$. Usually one can infer from the context which denotation is intended. — Bill Dubuque, Dec 16 '20 at 10:20
@BillDubuque: As I understand, pairwise coprime imply setwise coprime — Safwane, Dec 16 '20 at 10:23
Yes, generally if $,\color{#c00}{(a,b)=1},$ then $,(a,b,c,\ldots) =(\color{#c00}{(a,b)},c,\ldots) = (\color{#c00}1,c,\ldots) = 1,$ since gcd is associative. $ $ Or note $,(a,b,c,\ldots)\mid (a,b)\ [,!\color{#c00}{=!1}\ \text{here}]\ \ $ — Bill Dubuque, Dec 16 '20 at 10:30
Note that I refer to pairwise coprime (vs. $(A,B,C)= 1$) since your "i.e. there exist ..." makes it clear that is what you use in your question. What is you context? — Bill Dubuque, Dec 16 '20 at 10:36
Is that the entire exercise? If not, what is the rest of it? The more context you can give the better chance that you'll get an optimal answer for your context. — Bill Dubuque, Dec 16 '20 at 10:41

score 2 · Accepted Answer · answered Dec 16 '20 at 08:42

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

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

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

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