Multiplicative inverse example

Question

Theorem: Multiplicative inverse of x mod m is y with x*y = 1(mod m).

Example: For 4 modulo 7 inverse is 2: 2*4 = 8 = 1(mod 7).

Why 1(mod 7) is 8? I thought mod(1,7) = 1

$1\equiv8\pmod7$ means $7|1-8$. With mod as a binary operator (remainder after division), $\mod(1,7)= 1$ — J. W. Tanner, Feb 12 '20 at 02:56
$\large\begin{align} &8\equiv 1!!!!\pmod{!7}\ \ {\rm means}\ \ \ 7\mid 8-1,\ \ {\rm i.e.}\ \ 7 \ {\rm divides}\ 8-1\[.4em] &8\bmod 7, =, 1\ \ \ \ {\rm means}\ \ \ 8\div 7\ \ {\rm leaves\ remainder}\ = 1\end{align}$ $\ \ \ $ — Bill Dubuque, Feb 12 '20 at 03:28

score 1 · Accepted Answer · answered Feb 12 '20 at 03:08

There are two notions of mod, which are related but distinct.

One is a binary operation, which gives a result of two numbers, and is often used in computer science:

$\mod(a,b)$ is the remainder when $a$ is divided by $b$.

In that context, $\mod(1,7)=\color{blue}1$ because $1=0\times7+\color{blue}1.$

The other is a relation, often used in mathematics:

$a\equiv b\pmod n$ means $n$ divides $a-b$.

In that context, $1\equiv8\pmod7$, because $7\mid1-8$.

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