The definition of relative primality that I was taught was that:
Two numbers are relatively prime if the only common positive factor
of the two numbers is one.Every integer (except zero) divides zero and the only positive factor of one is one. Thus, the only common positive factor of zero and one is one.
Thus, it would seem that zero and one are relatively prime by the definition above. By convention is this not the case? i.e. zero is defined as not being relatively prime with any integer?
$\endgroup$ 43 Answers
$\begingroup$$\gcd(0,n)=n$ for all $n\in\mathbb{N}$. For $n=1$ it turns out to be $1$, so if you insist, $0$ and $1$ are relatively prime. Zero is not defined to be not relatively prime with any integer. It just so happens that it is divisible by any integer.
$\endgroup$ 1 $\begingroup$Every integer divides zero. The only integers that divide $1$ are $1$ and $-1$. The greatest common divisor of $0$ and $1$ is thus $1$. That makes them relatively prime.
$\endgroup$ $\begingroup$Yes, the largest integer that divides them both is $1$. In general $gcd (0,n)$ equals the absolute value of $n$ where $n$ is any non-zero integer.
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