Is there an intuitive way to tell if some integers are relatively prime?

Question

Suppose I have $a,b \in\mathbb Z$ such that $$\gcd(a,b)=1$$ My question is this:

Is there a way to intuitively know if $(a,b)$ are relatively prime without having to preform the Euclidean Algorithm and without knowing beforehand that $\gcd(a,b)=1$.

Answer 1

In some special cases, we do not need the Euclidean algorithm, for example when both numbers are even or divisble by $3$. Consecutive numbers can immediately be detected to be coprime. Possibly, we see immediately that one number is a multiple of the other.

Ignoring such or similar cases, without the Euclidean algorithm, we won't be able to distinguish between coprime numbers and numbers sharing only one large prime factor.

Answer 2

The other way would be prime decomposition of $a$ and $b$.

If they do not share any primes, they are relatively prime, otherwise $ \gcd(a,b)>1$