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. 

The problem is of the form $$ax\equiv\mod n.$$ If $\gcd(a,n)=1$, then the unique  solution is given by
$$x\equiv x_0\mod n$$ where $x_0$ is a particular solution of the given linear congruence. You may refer here. Note that $\gcd(19,35)=1.$ One such particular solution of $$19x\equiv 1\mod 35$$ is given $x_0=-11.$ Note that $-11\equiv 24\mod 35.$ The unique solution is given by$$x\equiv 24\mod 35.$$
