/sci/ - Science & Math

>>14629605
Why does every book define [math] \gcd[/math] as the common divisior that is divisible by all common divisors. Why not just call it, the common divisor that is the greatest? That's just putting cart before the horse. It should be treated as a theorem that the [math] \gcd[/math] is divisble by all other divisors, but of course every book leaves that out because that's just in muh definition, it cannot be proven. What is a good intuitive reasoning behind the [math] \gcd[/math] being divisible by all common divisors?