Question

Euclid's division lemma state that for any positive integers a and b, there exist unique
positive integers q and r such that a = bq + r where r must satisfy which of the following condition
​

Answer 1

If r must satisfy0≤r<b

Proof,

..,a−3b,a−2b,a−b,a,a+b,a+2b,a+3b,..

clearly it is an arithmetic progression with common difference b and it extends infinitely in both directions.

Let r be the smallest non-negative term of this arithmetic progression.Then,there exists a non-negative integer q such that,

a−bq=r

=>a=bq+r

As,r is the smallest non-negative integer satisfying the result.Therefore, 0≤r≤b

Thus, we have

$\bold{a=bq_1+r_1 , 0≤r_1 ≤b}$