Question

prove the euclid division
given and int j and k>0 then one can find a unique pair (q,r) of int state j=q*k +r and 0<r<k

Answer 1

This is a classical but rather lengthy proof that requires mathematical induction in three different cases.
One example proof can be found at:

http://www.proofwiki.org/wiki/Division_Algorithm

Post if you have further questions.