How do you get the multiplicative inverse by reading a multiplication table of modulo $n$?

Question

Here is the multiplication table for$\mod 7$: https://gyazo.com/82fb45bd89f61df3b44f00f67efc63c1

How do I read this to get the multiplicative inverse of something, for example like:

$5\mod 7$

I know the answer is $3$ by using a different method, but how do I get it using this table?

Answer 1

By definition, the multiplicative inverse of an element $x$ is the element $y$ that, when the two are multiplied, returns the identity.

To read a multiplication table (or generally a Cayley table to find any operative inverse) proceed along the row for your chosen element $x$ until you reach the identity. Proceed up that column and its label element will be the inverse $y$.