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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)

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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 $[a]\cdot[b]=[ab]$.