Question

definition of Euclid division Lemma
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Answer 1

Answer:

According to Euclid’s Division Lemma if we have two positive integers a and b, then there exist unique integers q and r which satisfies the condition a = bq + r where 0 ≤ r ≤ b.

The basis of the Euclidean division algorithm is Euclid’s division lemma. To calculate the Highest Common Factor (HCF) of two positive integers a and b we use Euclid’s division algorithm. HCF is the largest number which exactly divides two or more positive integers. By exactly we mean that on dividing both the integers a and b the remainder is zero.

Answer 2

In number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely:[note 1]

Euclid's lemma — If a prime p divides the product ab of two integers a and b, then p must divide at least one of those integers a and b.

For example, if p = 19, a = 133, b = 143, then ab = 133 × 143 = 19019, and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as well. In fact, 133 = 19 × 7.

Inherently, if the premise of the lemma does not hold, i.e., p is a composite number, its consequent may be either true or false. For example, in the case of p = 10, a = 4, b = 15, composite number 10 divides ab = 4 × 15 = 60, but 10 divides neither 4 nor 15.

This property is the key in the proof of the fundamental theorem of arithmeticIt is used to define prime elements, a generalization of prime numbers to arbitrary commutative rings. Euclid's Lemma shows that in the integers irreducible elements are also prime elements. The proof uses induction so it does not apply to all integral domains.