Editing Coprime integers (section)

{{Short description|Two numbers without shared prime factors}}

In [[number theory]], two [[integer]]s {{mvar|a}} and {{mvar|b}} are '''coprime''', '''relatively prime''' or '''mutually prime''' if the only positive integer that is a [[divisor]] of both of them is 1.<ref>{{cite book |last1=Eaton |first1=James S. |title=A Treatise on Arithmetic |date=1872 |publisher=Thompson, Bigelow & Brown |location=Boston |page=49 |url=https://archive.org/details/atreatiseonarit05eatogoog |access-date=10 January 2022 |quote=Two numbers are ''mutually'' prime when no whole number but ''one'' will divide each of them}}</ref> Consequently, any [[prime number]] that divides {{mvar|a}} does not divide {{mvar|b}}, and vice versa. This is equivalent to their [[greatest common divisor]] (GCD) being 1.<ref>{{harvnb|Hardy|Wright|2008|loc=p. 6}}</ref> One says also {{mvar|a}} ''is prime to'' {{mvar|b}} or {{mvar|a}} ''is coprime with'' {{mvar|b}}.

The numbers 8 and 9 are coprime, despite the fact that neither—considered individually—is a prime number, since 1 is their only  common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a [[reduced fraction]] are coprime, by definition.