modular arithmetic multiplicative inverse?

Question

Can someone please provide some examples of finding multiplicative inverse in modular arithmetic. then solve giving an explanation.

for example:
what is the multiplicative inverse of mod7?
I think its 8 but I am not sure why. what about others. Thanks for your help.

Answer 1

Inverses multiply to 1.



Mod 7:

It cannot be 8, the only choices available are 0 to 6.



0 does not have an inverse.

1 is always its own inverse.

2 and 4 are inverses since 2 x 4 = 8 = 1 mod 7.

3 and 5 are inverses since 3 x 5 = 15 = 1 mod 7.

6 is its own, since 6 x 6 = 36 = 1 mod mod 7.



Example problem:

What is the inverse of 9 (mod 11) ?



We want 9 * x = 1 (mod 11)

or 9x = 11 y + 1

multiples of 11 are 11, 22, 33, 44 ... and 44 is one less than a multiple of 9.

So the inverse of 9 (mod 11) = 5

5 * 9 = 45 = 1 mod 11.



Equations of the form 9x = 11 y + 1

to be solved in integers are called Diophantine Equations,

and there are systematic ways of solving them,

which, not too suprisingly, involve the use of modular arithmetic.

Answer 2

Inverse Modulo

Answer 3

A has a multiplicative inverse modulo B implies that there is C,D such that AC=DB+1 Hence DB=-AC+1 =1 (mod A) So D is the inverse modulo A of B