Wednesday, February 13, 2019

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 $0\leq r \leq b$.



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