Question 105
For every integer $n \geqslant 1$, let $f_{n}$ denote the sum of all the remainders obtained when dividing $n$ by the numbers $1,2, \ldots, n$. For example, dividing $5$ by $1,2,3,4$ and $5$ gives the remainders $0,1,2,1$ and $0$, so that $f_{5}=0+1+2+1+0=4$.
Find all integers $n \geqslant 2$ such that $f_{n}=f_{n-1}+n-2$.
Find all integers $n \geqslant 2$ such that $f_{n}=f_{n-1}+n-2$.