Q: What is a "normal" multiplicative inverse? A: it's the thing you multiply by to get 1.

That's why the multiplicative inverse of 3 is 1/3 - because 3(1/3) = 1.

It's the same thing in modular arithmetic, except now it works like this:

Q: What do you have to multiply 3 by to get 1, MODULO 7? A: 5

That's because (3)(5) = 15 = 1 + 14 = 1 MODULO 7.

Thus the multiplicative inverse of 3 is 5, when working mod 7.

0 never has a multiplicative inverse. In modular arithmetic as well as regular arithmetic, you can never divide by 0.

Other examples:

(1)(1) = 1 mod 5

(2)(3) = 1 mod 5

(4)(4) = 1 mod 5

(1)(1) = 1 mod 6

(5)(5) = 1 mod 6

2, 3, and 4 have no multiplicative inverse mod 6.

(1)(1) = 1 mod 7

(2)(4) = 1 mod 7

(3)(5) = 1 mod 7

(6)(6) = 1 mod 7

(1)(1) = 1 mod 8

(3)(3) = 1 mod 8

(5)(5) = 1 mod 8

(7)(7) = 1 mod 8

2, 4, and 6 have no multiplicative inverse mod 8.

(1)(1) = 1 mod 9

(2)(5) = 1 mod 9

(4)(7) = 1 mod 9

(8)(8) = 1 mod 9

3 and 6 have no multiplicative inverse mod 9.