Is every positive integer relatively prime to $1$?

Question

Because $\gcd(k, 1) = 1$ where $k > 0$ is an integer.

Answer 1

Yes. $1$ is relatively prime to every positive integer, including itself. $1$ is also relatively prime to every negative integer, and to $0$.

There are many definitions of relatively prime which you can use to check this, including the one you cite. For integers $a$ and $b$, the following are all equivalent:

• $a$ and $b$ are relatively prime;

• $\gcd(a,b) = 1$;

• No prime $p$ divides both $a$ and $b$;

• There exists integers $x$ and $y$ such that $ax + by = 1$;

• $a\mathbb{Z} + b\mathbb{Z} = \mathbb{Z}$ (this is addition of ideals);

• $\text{lcm}(a,b) = ab$.

Answer 2

What's your definition of "relatively prime"? For me, I say $a$ and $b$ are relatively prime if there is no prime number which divides both of them simultaneously.

So if I were thinking about $k$ and $1$, since no prime number divides $1$ in the first place, then no prime number can divide them both simultaneously, and so using my definition, $k$ and $1$ are relatively prime.