Let $n = pq$, where $p = 13710221545914561761$ and $q = 11066328760152681859$ are both primes. Find the (two) square roots of $1 \mod n$ that are not $\pm1\mod n$. Let the square roots be $a_1\mod n$ and $a_2 \mod n$, with $a_1,a_2$ element of $\{0,n-1\}$ and $a_1 > a_2$, find $a_1$.
Solution. Begin with the congruence with the largest modulus, Rewrite this congruence as an equivalent equation: $$x=13710221545914561761 j + 1$$
Substitute this expression for $x$ into the congruence with the next largest modulus:
$$x\equiv 1\mod 11066328760152681859$$ $$\Rightarrow 13710221545914561761 j + 1 \equiv 1\mod 11066328760152681859$$