If dividing $n$ by $m$ yields remainder $r$, then dividing $-n$ by $m$ yields remainder $-r$

Question

Let m and n be positive integers and let r be the nonzero remainder when n is divided by m. Prove that when -n is divided by m, the remainder is m - r

So far I've tried I get

n = qm + r and -n = q'm + r'

solving for r and r' i get

r = n -qm and r' = n - q'm

plugging r = n - qm back into r' = m - r i get

r' = m - n + qm

reordering gives back -n = (-1-q)m + r'

similarly the plugging in the r' = -n - q'm into r' = m - r

which gives back n = (-1-q')m + r

Not sure what to do next or if I'm doing it correctly

Answer 1

Suppose $n=qm+r$ s.t. $0<r<m$ thus $-n=-qm-r=-(q+1)m+(m-r)$ and $0<m-r<m$.