fermat's little theorem and residue classes 


I am trying to understand fermat's little theorem in residue classes but the below slides make absolutely no sense to me. In computer classes a' means if you have 3 then 3' would be 6 because 3+6=9 so I am really confused here about what they are doing..
I know that residue classes mod m basically means the remainder when mod by m but I dont really understand what they mean or do by [a][b]=[a'][b']
Also, really not sure how they are finding the inverse...the general equation is a congruent to b (mod m)
 A: The primes in the statement of prop 4.2.2 are not operators; $a'$ and $b'$ are simply names of variables which are different from $a$ and $b$, although suggestively named.
It is, however, not very clearly written. It appears that the author is using $[a]$ and $[a']$ denote the residue classes of the integers $a$ and $a'$, and so forth. But it is strictly speaking nonsense to write "if $[a]=[a']$ and $[b]=[b']$ modulo $m$ ...". For then $[a]=[a']$ simply asserts that the residue classes are the same, and this identity is just an identity between sets of numbers; there is nothing modular about the way these sets are equal.
But if the author does mean the premises to be $[a]=[a']$ and $[b]=[b']$, then the conclusions $[a]+[b]=[a']+[b']$ and $[a][b]=[a'][b']$ are completely vacuous, because of course we're allowed to substitute equals for equals.
What the proposition ought to have been, in order to be meaningful, is

If $a\equiv a'\pmod m$ and $b\equiv b'\pmod m$, then $(a+b)\equiv(a'+b')\pmod m$ and $ab\equiv a'b' \pmod m$.

or, equivalently,

If $[a]=[a']$ and $[b]=[b']$, then $[a+b]=[a'+b']$ and $[ab]=[a'b']$.

... and because of this fact it is possible and meaningful to define the sum and product of residue classes by $[a]+[b]=[a+b]$ and $[a]\cdot[b]=[ab]$.
