1=ax+by, for all integers x and y 1=ax+by, for some integers x and y 1=ax+by,...

Step 1
Two integers are relatively prime when there are no common factors other than 1. This means that no other integer could divide both numbers.
Two integers a,b are called relatively prime to each other if ${\displaystyle gcd(a,b)=1}$.
Step 2
For any positive integers a and b, there exist integers x andy such that ${\displaystyle ax+by=gcd(a,b)}$. Furthermore, as x and y vary over all integers a${\displaystyle x+by}$ attains all multiples and only multiples of gcd(a, b).
Here ${\displaystyle gcd(a,b)=1}$
So there exist integers x and y such that ${\displaystyle ax+by=1}$
So the correct choice is b.
${\displaystyle 1=ax+by}$, for some integers x and y.