elementary number theory - What is the correct way of expressiong remainder in Mathematics

$45/7 \quad$ remainder $=3$

What is the correct way of representing this mathematically? I am asking this question because in this site, many times, experts use different ways to denote remainders. I am giving it below

(a) $45$ mod $7$ $=3$

(b) $45$ mod $7$ $\equiv3$

(c) $45\%7$ $=3$ (I believe this is mostly for programming and cannot generally use for mathematics. there is a thread for it)

(d) $45\equiv 3\pmod 7$

It is true that we can easily understand from the last expression that $45$ divided by $7$ gives $3$ as remainder. But, this relation is actually used to tell $45$ and $3$ gives same remainder when divided with $7$.

So, my understanding is that we can only (a). Please tell if I am right or wrong.

Answer

To capture the nature of division of a number $a$ by another number $b$ (which seems to be what you're trying to convey in $a$, we can write $$a = qb + r$$ where q represents the unique quotient, and $r$ ($0\leq r\lt b$) represents the unique remainder.

We can also write $$a \equiv r \pmod b$$

The notation of the second form does not necessarily require that the '$r$' be such that $0\leq r \leq b$.