Prove gcd or disprove

Question

I want to know how to prove or disprove:

1. For all integers $a, b$, we have $\text{gcd}(a,b)\le\min(a,b)$
2. For all integers $a, b$, we have $\text{gcd}(a,b)\le\min(|a|,|b|)$
3. For all non-zero integers $a, b$, we have $\text{gcd}(a,b)\le\min(|a|,|b|)$.

This is everything I want to know. Thank you.