I am in a number theory class which does not have a text and the following definition of REM is giving me trouble. That is,
"We will denote by $aREMm$ the remainder left when $a$ is divided by $m$ (i.e $r$)."
I interpreted this as saying if $a=mq+r$, then $r=aREMm=a-mq.$
However, when trying to use this for the following proof things begin to get problematic.
We want to prove the following: Let $a,b,$ and $m$ be integers such that $a\equiv b(mod\space m)$. Then, $a\equiv b REMm(mod\space m).$
Writing $a\equiv b REMm(mod\space m)$ makes me almost cringe because I feel like I have no idea what that means, but I will try to write it down algebraically from definition.
$a\equiv b REMm(mod\space m)$ means that $m|a-(bREMm)$, or in otherwords $a=mk+bREMm$. However, $bREMm$ is the remainder left when $b$ is divided by $m$, so we have $bREMm=r=b-mz$. So, ultimately what I see the question asking is
If $m|a-b$, then $m|a-(b-mz).$
Is this a correct interpretation of the statement?
Any help would be greatly appreciated. Thank you.