Hi,

**problem: ** .

If

and

are in

, let

be the least positive remainder obtained by dividing the (ordinary) sum of

and

by

,

and similarly, let

be the least positive remainder obtained by dividing the (ordinary) product of

and

by

Prove that if

is a field, then either the result of repeatedly adding

to itself is always different from

,

or else the first time that it is equal to

occurs when the number of summands is a prime.

**attempt:**
I look at

where

is the number of times 1 is added to itself.

,

is prime since

is a field.

when

is a multiple of

,

.

For

,

.

For

,

I am really bad at this so any suggestion is very welcome!

Thanks