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If we have $a \equiv b \pmod{n}$ then $a$ and $b$ are congruent to each other modulo $n$, correct?

What do we "call" $a$ and $b$? Because sometimes these numbers can be negative. Would they be remainders? Residues? Do we say these remainders (residues?) belong to the same "congruence class mod $n$"? Are remainders and residues synonymous?

Trying to get the terminology / lingo right here. When do we use which words? How do we describe what all this notation is representing / saying?

Ethan Bolker
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Sean Hill
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    Yes; see congruence relation : "two numbers $a$ and $b$ are said to be congruent modulo $n$". $a$ and $b$ are numbers and $n$ is called the modulus of the congruence. – Mauro ALLEGRANZA Oct 31 '18 at 13:35
  • Are $a$ and $b$ remainders? Residues? – Sean Hill Oct 31 '18 at 13:37
  • "Like any congruence relation, congruence modulo $n$ is an equivalence relation, and the equivalence class of the integer $a$, denoted by $a_n$, is the set ${ \ldots, a − 2n, a − n, a, a + n, a + 2n, \ldots }$. This set, consisting of the integers congruent to a modulo $n$, is called the congruence class or residue class or simply residue of the integer $a$, modulo $n$." – Mauro ALLEGRANZA Oct 31 '18 at 13:38
  • Thus, residue of $a$ modulo $n$ is the set of all integers congruent to $a$ modulo $n$. – Mauro ALLEGRANZA Oct 31 '18 at 13:40
  • Huh for some reason I thought residue meant like remainder or a single number and not a set of integers, interesting. Is there a word for a given integer taken from the residue of $a$ mod $n$? – Sean Hill Oct 31 '18 at 13:44
  • reminder is pertinent because $a$ and $b$ are congruent modulo $n$ exactly when they have the same reminder when divided by $n$. Thus, the conguence class (or residue) contains all numbers that have the same reminder. – Mauro ALLEGRANZA Oct 31 '18 at 13:50
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    @Sean "residue" is an overloaded (ancient) term. It can mean either a representative of a congruence class or the entire class. It can also mean "congruent", e.g. older texts often write "$b$ is a residue of $a$ modulo $n$" for $a\equiv b\pmod{n}.,$ It predates modern (abstract) algebra so one can't expect it to be completely consistent with modern syntax and semantics. – Bill Dubuque Oct 31 '18 at 14:02
  • @MauroALLEGRANZA Does it make sense to say that a negative number can have a positive/negative remainder? For example $-10$ divided by $7$ might have remainder $-3$ or remainder $4$? – Sean Hill Oct 31 '18 at 14:06
  • @Sean remainders typically denote normal / canonical reps, e.g. least nonnegative or least magnitude (signed). reps, e.g. $\bmod 5$ they are ${0, 1, 2, 3, 4}$ or ${ -2, -1, 0, 1, 2 }. $ More generally we can use any complete systems of incongruent reps – Bill Dubuque Oct 31 '18 at 14:08
  • We have that $-10$ divided by $7$ gives a quotient $q$ and a reminder $r$. The rule is : $7 \times q+r=-10$. – Mauro ALLEGRANZA Oct 31 '18 at 14:10
  • See the post : negative number divided by positive number, what would be remainder? – Mauro ALLEGRANZA Oct 31 '18 at 14:11
  • And see modulo operation. – Mauro ALLEGRANZA Oct 31 '18 at 14:12

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