DoorstepTutor (NEET, JEE, IMO, NSO, NTSE...)'s video: Euclid s Division Lemma- Its Scope and Proof Visualizing Division Lemma Working Several Examples

What is Euclid's Division Lemma- Scope, Meaning and Application using One Simple Example Given integers a and b, with b not equal to 0, there exist unique integers q and r satisfying a = qb + r (0 less than equal r less than |b|). The integers q and r are called, respectively, the quotient and remainder in the division of a by b. 1. The lemma is valid for all a and b which are integers such than b not equal to zero. 2. q and r are unique. 3. r can be zero or greater than zero but has to be less than absolute value of b. 4. b can be bigger than a (for example, if a and b are also positive q would be equal to zero and r = a). How to Visualize and Understand the Euclid's Division Lemma- For Both Positive and Negative Divisor Simple visualization to easily understand Euclid's Division Lemma? Simple example with positive divisor and then extended to negative divisor as well. Proof of Euclid's Division Lemma- Existence/Uniqueness of q and r, 0≤r≤b, +ve & -ve Divisor/Dividend Division Lemma: For each pair of positive integers a and b, we can find unique integers q and r satisfying the relation a = bq + r , 0 ≤ r ≤ b. Here we will attempt to prove the lemma over entire range of integers (divisor not equal to zero).