Coprime integers: $\gcd (a,b)=1$

Integers $a$ and $b$ are coprime, or relatively prime, when

$\gcd (a,b)=1$

(Common divisor, and the greatest common divisor $\gcd (a,b)$, with the convention $\gcd (0,0):=0$). The relation is symmetric, since $\gcd (a,b)=\gcd (b,a)$ ($\gcd$ is symmetric and unchanged by signs: $\gcd (a,b)=\gcd (b,a)=\gcd (\mathrm{\mid }a\mathrm{\mid },\mathrm{\mid }b\mathrm{\mid })$; moreover $\gcd (a,0)=\mathrm{\mid }a\mathrm{\mid }$, $\gcd (a,1)=1$, $\gcd (a,a)=\mathrm{\mid }a\mathrm{\mid }$, and $\gcd (a,b)\ge 1$ unless $a=b=0$), and unchanged by signs, since $\gcd (a,b)=\gcd (\mathrm{\mid }a\mathrm{\mid },\mathrm{\mid }b\mathrm{\mid })$.

Boundary values. $\mathbb{N}$ contains $0$ and $\mathbb{Z}$ contains $0$ and $1$, so the three degenerate pairs are recorded explicitly, each read off $\gcd$ is symmetric and unchanged by signs: $\gcd (a,b)=\gcd (b,a)=\gcd (\mathrm{\mid }a\mathrm{\mid },\mathrm{\mid }b\mathrm{\mid })$; moreover $\gcd (a,0)=\mathrm{\mid }a\mathrm{\mid }$, $\gcd (a,1)=1$, $\gcd (a,a)=\mathrm{\mid }a\mathrm{\mid }$, and $\gcd (a,b)\ge 1$ unless $a=b=0$:

• $a$ and $1$ are coprime for every integer $a$, since $\gcd (a,1)=1$; in particular $0$ and $1$ are coprime.
• $0$ and $b$ are coprime exactly when $b=1$ or $b=-1$. Indeed $\gcd (0,b)=\gcd (b,0)=\mathrm{\mid }b\mathrm{\mid }$, so coprimality says $\mathrm{\mid }b\mathrm{\mid }=1$, and by Absolute value in $\mathbb{Z}$: $\mathrm{\mid }a\mathrm{\mid }\ge 0$; $\mathrm{\mid }a\mathrm{\mid }=0$ exactly when $a=0$; $\mathrm{\mid }-a\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }$; $\mathrm{\mid }ab\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }$; $-\mathrm{\mid }a\mathrm{\mid }\le a\le \mathrm{\mid }a\mathrm{\mid }$; and $\mathrm{\mid }a\mathrm{\mid }\le c$ exactly when $-c\le a\le c$ together with the case split of The absolute value $\mathrm{\mid }a\mathrm{\mid }$ of an integer this holds exactly for $b=1$ and $b=-1$, the two units of $\mathbb{Z}$ ($(\mathbb{Z},\cdot ,1)$ is a commutative monoid whose group of units is $\{1,-1\}$; equivalently $u\mid 1$ holds exactly for $u=1$ and $u=-1$).
• $0$ and $0$ are not coprime, since $\gcd (0,0)=0$ by the convention of Common divisor, and the greatest common divisor $\gcd (a,b)$, with the convention $\gcd (0,0):=0$, and $0\mathrm{\ne }1$: if $1$ were $0$ then $-1=-0=0=1$, contradicting the two units being distinct ($(\mathbb{Z},\cdot ,1)$ is a commutative monoid whose group of units is $\{1,-1\}$; equivalently $u\mid 1$ holds exactly for $u=1$ and $u=-1$).

• Coprime is a statement about the pair, not about either integer. Neither $a$ nor $b$ need be prime, and primality is not defined anywhere on this page: $4$ and $9$ are coprime and neither is prime. What coprimality says is that the only common divisors are the units $1$ and $-1$ ($a$ and $b$ are coprime if and only if $ax+by=1$ for some integers $x,y$; and in that case the only common divisors of $a$ and $b$ are $1$ and $-1$).

• Why it is worth a name before primes appear. The lemma that carries the weight in elementary number theory — if $a\mid bc$ and $a$ is coprime to $b$ then $a\mid c$ (If $\gcd (a,b)=1$ and $a\mid bc$ then $a\mid c$; and if $a\mid c$, $b\mid c$ and $\gcd (a,b)=1$ then $ab\mid c$) — needs coprimality and no primality at all. Primes enter later, and their key property is a special case of that lemma rather than an independent fact.