Let $\gcd (a, b) > 1$. Show that the equation $ax + by= 1$ has no solutions.
We'll use Proposition (1.17)
The Diophantine equation $ax + by= c$ has integer solutions if and only if $\gcd (a, b) \mid c$.
By Proposition 1.17 the equation $ax + by =1$ has integer solutions if $ \gcd(a,b) \mid 1$. But since we're given $\gcd(a,b)>1$ this is not possible. A number larger than 1 can't divide 1.
And so $ax + by= 1$ has no (integer) solutions.