I think I am not understanding something properly as I keep having problems with modular arithmetic for some reason. For example, I am given this:

Compute:

mod 71

So I understand that there is a number x that is =

less a multiple of 71. So with my calculator I do the following:

the original statement is congruent with 32 mod 71

Or I just find how many whole times 71 will go into

. multiply 71 by that number and then subtract it form

to be left with 32.

I can't seem to understand my notes on how to do it without a calculator and now I have a problem doing the following (my calculator just shows error):

mod 89

Can someone walk me through what they are thinking/trying to do when breaking these down without a calculator? or direct me to a resource that does a better job of explaining this - I can't seem to grasp it yet.

Thanks.