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Let $a$, $b$, and $n$ be integers with $0 \le a,b \lt n$. If $\,a \equiv b \pmod n,\,$ i.e. $\,a-b = pk\,$ for an integer $k,\,$ then $a=b$. Why does this congruence imply the equality?

Bill Dubuque
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It is not enough that $a,b$ are nonnegative, it is also necessary that $a,b$ are less than $n$. With both of these conditions, then $a\equiv b$ implies that $a=b$. The reason is that $a\equiv b$ means that $a-b$ is an integer multiple of $n$. However, this integer multiple cannot be $n$ or greater (else $a\ge b+n$, impossible), nor $-n$ or smaller (else $a+n\le b$, also impossible). Hence the only integer multiple of $n$ available is $0n$.

vadim123
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    You might want to specify that it is impossible because $a \ge b+n > b$ and also $a \le b-n < 0$, violating the conditions. – Kenny Lau Aug 16 '17 at 03:05
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Hint: their $\rm\color{#c00}{difference}$ is a natural smaller than $\,n\,$ and divisible by $\,n\,$ so it must be $\,0,\,$ i.e.

w.l.o.g. $\ 0 \le b\le a < n\,\overset{\rm subtract\ b}\Longrightarrow\, 0\le \color{#c00}{a\!-\!b <} n\!-\!b\le\color{#0a0} n \ $ so $\ \color{#0a0}n\mid \color{#c00}{a\!-\!b}\iff a\!-\!b = 0$

Remark $ $ This shows the uniqueness part of the fact that $\,0,1,2,\ldots,n\!-\!1\,$ form a complete system of representatives for equivalence classes of $\,\Bbb Z_n = $ integers $\!\bmod n,\,$ or, equivalently, the uniqueness of the remainder in the (standard) division algorithm. The same proof works for any sequence of $n$ consecutive integers since - as above - any two elements differ by less than the extreme elements, so less than $n,\,$ so their difference is divisible by $n$ only if they are equal. An example of an alternative sequence of remainders is those having least magnitude, e.g. $\,0,\pm1, \pm2\pmod{\!5}$. The argument easily generalizes to any arithmetic progression whose step-size is coprime to the modulus.

Corollary $\ A\equiv B\pmod{\!n}\iff \underbrace{A\bmod n = B\bmod n}_{\textstyle \color{#c00}{a = b}}$

J. W. Tanner
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Bill Dubuque
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  • How can we show that if $a-b\leq n$ and $n\mid a-b$ then $a-b=0$? That part struggles me :/ – manooooh Dec 14 '19 at 22:09
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    @man An $n$-multiple $,kn\neq 0,$ has greater magnitude $,|kn| = |k||n|\ge |n|,$ by $|k|\ge 1,$ by $,k\neq 0\ \ $ – Bill Dubuque Dec 14 '19 at 22:26
  • Sorry I do not understand what can you conclude from there. If $k\neq0$ and $kn\neq0$ then $n=0$? – manooooh Dec 14 '19 at 22:29
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    @man That $m=0$ is the only multiple $m$ of $n$ with $,0\le \color{#c00}m <\color{#0a0} n,$ since all other multiples $>0,$ are $\ge n.\ $ Explicitly: the natural multiples of $n$ are $,0,n,2n,3n,\ldots$ and there are $\rm\color{#c00}{none}$ strictly between $0$ and $n.\ $ Said more informally, an integer divides a smaller integer only when the smaller $= 0\ $ – Bill Dubuque Dec 14 '19 at 22:32
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    Unique remainder $\Rightarrow$ unique quotient by $,qn+r=q′n+r\Rightarrow qn=q′n\overset{n\neq 0}\Rightarrow q=q′\ $ – Bill Dubuque Apr 30 '23 at 23:56
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The key trick here is the interplay between magnitude and divisibility — every nonzero multiple of $n$ must have a magnitude at least as large as $n$.

Symbolically, if $k \neq 0$, then $|kn| \geq |n|$.

Since integers in the same congruence class are separated by multiples of $n$, this puts a limit on how close they can be. We have

If $a \equiv b \bmod n$ and $a \neq b$, then $|a-b| \geq |n|$

The premise of the problem constrains both $a$ and $b$ to a narrow interval: they are both contained in the set $\{ 0, 1, \ldots, n-1 \}$. The largest possible difference between any two elements of this set has magnitude $n-1$. That is, for any two elements $x,y$ of this set, we have $|x-y| \leq n-1$.

So by the previous observation, the fact that $a \equiv b \bmod n$ implies that one of the following is true:

  • $a = b$
  • $|a-b| \geq n$

and we can rule out the second possibility because $|a-b| \leq n-1$.