least residue of 100! + 102! mod 101

Question

Here is what I have:

want least $x$ s.t. $x \equiv_{101} 100! + 102!$

by Wilsons theorem

$100! \equiv_{101} -1$ **

and we know

$102 \equiv_{101} 1$

I was thinking:

$x \equiv_{101} 100!(1 + 101 \cdot 102)$

then $102$ becomes $1$? and again $102$ becomes $1$ hence

$x \equiv_{101} 100!$ and from ** we get $x = -1$

I KNOW THE LEAST RESIDUE IS 1, how do I go from the residue I got which is -1 to the LEAST residue?

Answer 1

$100!+102!\equiv -1+0\equiv -1\equiv 100\bmod 101$