Question

State “Euclid’s Division Lemma”?

Answer 1

Hint- Here, we will proceed by stating the Euclid’s Division Lemma and then giving the general form of the representation of any two positive integers. Then, after that we will be discussing a simple example related to Euclid’s Division Lemma.

Complete step-by-step answer:
Euclid’s Division Lemma is a proven statement which is used to prove other statements in the branch of mathematics. It is the basis for Euclid’s division algorithm.
Statement of Euclid’s division lemma: Euclid’s division lemma states that, if there are two positive integers i.e., a and b, then there exists unique integers i.e., p and q such that these integers satisfies the condition a = bp +q where $0 \leqslant q \leqslant b$
Let us take an example of the division of positive integers by a positive integer, say 52 by 7. In this particular example, 52 is the dividend, 7 is the divisor. When 52 is divided by 7, the quotient obtained will be 7 and the remainder left will be 3. We can write the result in the following form by applying the Euclid’s Division Lemma method, $52 = \left( {7 \times 7} \right) + 3;0 \leqslant 3 \leqslant 7$

Note- Euclid’s Division Lemma method is used to calculate the highest common factor (HCF) of any two positive integers where the highest common factor (HCF) is the largest number which exactly divides the two given positive integers (i.e., when HCF is divided by both the numbers it gives zero as the remainder).