Solution of $19 x \equiv 1 \pmod{35}$ $19 x \equiv 1 \pmod{35}$
For this, may I know how to get the smallest value of $x$. I know that there is a theorem like $19^{34} = 1 \pmod {35}$. But I don't think it is the smallest. 

Since, $\gcd(19,35)=1\implies \exists x_0,y_0\in \Bbb Z$ such that $19x_0+35y_0=1$.
You can apply euclid's algorithm, $$35=1\cdot19+16$$ $$19=1\cdot16+3$$ $$16=5\cdot3+1$$ $$\implies 1=16-5\cdot3=16-(19-1\cdot16)5=(35-19)-5(19-(35-19))=6\cdot35-11\cdot19$$ $$\implies19\cdot11=35\cdot6-1\implies 19\cdot(-11)\equiv 1\pmod {35}\implies 19\cdot24\equiv 1\pmod {35}$$
