A multiplicative inverse of 5 module 7 is?

Answer 1

A multiplicative inverse of 5 mod7 would be a number n ( not a unique one) such that 5n=1

Let's look at the possible numbers

5x1=5mode 7
5x2=10=3 mod 7
5x3=15=1 mod 7 THAT WILL DO IT

3 is the multiplicative inverse of 5 mod 7.

What about the others? 5x4=20, that is -1 mod 7 or 6
5x5=25 which is 4 mod 7
5x6=30 which is -5 or 2 mod 7

How did we know it existed? Because 7 is a prime. For every Prime number p and positive integer n, there exists a finite field with pn elements. This is an important theorem in abstract algebra. Since it is a field, it must have a multiplicative inverse. So the numbers mod 7 make up a field and hence have a multiplicative inverse.