A number theory problem by Hamza A

The probability that any number is divisible by a prime (or in fact any integer) P is $\frac{1}{P}$ , for example, every 5th integer is divisible by 5. Hence the probability that n numbers are divisible by P is $\frac{1}{P^n}$ , and the probability that n numbers are coprimes is $1-\frac{1}{P^n}=1-P^{-n}$ . Any finite collection of divisibility events associated to distinct primes is mutually independent so the probability that n numbers are coprime is given by a product over all primes $\prod_{primes} 1-P^{-n} = \left[\prod_{primes} \frac{1}{1-P^{-n}}\right]^{-1}$ And by the Euler product formula $\left[\prod_{primes} \frac{1}{1-P^{-n}}\right]^{-1}=\zeta(n)^{-1}$ now for n=3 the probability is equal to $\frac{1}{\zeta(3)}$

What the problem states is that we choose randomly 3 positive integers (a,b,c), we know that 3 numbers will be relatively prime if and only if they have a greatest common divisor of 1. By definition the probability that n randomly chosen positive integers, are relatively prime is $[\zeta({3})]^{-1}$ look at https://mathworld.wolfram.com/RelativelyPrime.html