The statement of Division Algorithm is, Given integers a and d, d =/= 0, there exist unique integers, q and r, such that a = qd + r,with 0 <= r < |d| We can state it in this way, If a and d are integers such that d =/= 0 THEN, there exist unique integers, q and r, such that a = qd + r,with 0 <= r < |d| Now,suppose we have integers 'a' of this form, a=2q+1 where a and q are integers. Now,I have seen written in my textbook that if we have integers of this form then it is not divisible by '2' as it leaves remainder '1'. What I DO NOT understand is that how do we KNOW that 1 is remainder? I think it uses DIVISION LEMMA. By looking at "a=2q+1" we can only determine two things. First,that we have two integers q and 1 such that 0<=1<2.(similar to
the conclusion of the division algorithm) second,2q+1=a BUT how can this INFORMATION (.....two integers q and 1(r) such that 0<=1<2 AND 2q+1=a) MAKE us say that '1' is remainder? HOW DO WE APPLY THE DIVISION ALGORITHM TO CLAIM THAT '1' IS REMAINDER? (****I KNOW THAT WE CAN APPLY A THEOREM ONLY IF THE "REQUIRED CONDITION" or THE CONDITION WHICH IS MENTIONED AFTER 'IF.......' IS TRUE****) Please Mention the steps you follow for applying the theorem.
"Unique" means there is only one possible way of doing it.
a=2n+1 is one way of doing it.
So a=2n+1 is the only way of doing it.
So if a=2q+r is the unique way of doing it. And a=2n+1 is the only way of doing it. Then q=n and r=1.
There is simply no other possible interpretation.