Solve Modular Arithmetic Homework: 12^9 mod71

[sara_87]

Homework Statement

find: 12^9 mod71

Homework Equations

The Attempt at a Solution

=12(12^8) mod71
= 12mod71 x 12^8mod71
= 12 x (12^2)^4mod 71

Now I'm stuck. My teacher solved it but i don't understand what he did so can someone explain how to do it in a very basic way.?
Thank you v much.

 

[Dick]

12^2 mod 71=2 mod 71. So 12^8 mod 71=2^4 mod 71.

 

[sara_87]

how come 12^2mod71=2mod71 ??

 

[Dick]

Because 12^2=144. 144 mod 71=2 mod 71.