I’m self-studying number theory using George Andrew’s textbook. I’m at the chapter of congruence modulo. There is one or two parts that I couldn’t quite figure out. Wonder if someone could point things out for me.
By definition, if $c≠0$, $a≡b \pmod c$ provided that $(a-b)/c$ is an integer. That is $c \mid (a-b)$.
If $a= 5$, $b=-3$, $c=8$
5 is congruent to -3 modulo 8, $5≡-3 \pmod 8$ since $(5-(-3))/8$ is an integer of $1$.
I also read on Wikipedia that "Equivalently, a ≡ b mod n can also be thought of as asserting that the remainders of the division of both a and b by n are the same." https://en.wikipedia.org/wiki/Modular_arithmetic
If that’s the case, using the same example. The remainder of (5/8) is 5, while the remainder of (-3/8) is -3. They are not the same.
I think I’m missing a point here. Could someone please help point out where my reasoning is wrong?
I quote the definition of Euclidean division for integers given in Wikipedia: "Given two integers $a$ and $b$, with $b\neq 0$, there exist unique integers $q$ and $r$ such that $a= bq + r$ and $0\le r < |b|$, where $|b|$ denotes the absolute value of $b$".
In the example we have $5=8.0+5$ and $-3=8.(-1)+5$, so the remainders are the same.