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Introduction
Euclid's Division Lemma is a fundamental concept in number theory that helps in understanding the properties of division. It states that for any two positive integers 'a' and 'b', there exist unique integers 'q' and 'r' such that 'a = bq + r', where 'q' is the quotient and 'r' is the remainder.

Explanation
Euclid's Division Lemma can be better understood through the following steps:

Step 1: Statement of Euclid's Division Lemma
Euclid's Division Lemma states that for any two positive integers 'a' and 'b', where 'a' is the dividend and 'b' is the divisor, there exist unique integers 'q' and 'r' such that 'a = bq + r', where 'q' is the quotient and 'r' is the remainder. Additionally, the remainder 'r' will always be less than the divisor 'b'.

Step 2: Understanding the Quotient and Remainder
The quotient 'q' represents the number of times the divisor 'b' completely divides the dividend 'a'. It can be obtained by dividing 'a' by 'b'. The remainder 'r' is the remaining part after the division, which is always less than 'b'.

Step 3: Examples
Let's consider an example to illustrate Euclid's Division Lemma. Suppose we have 'a = 17' and 'b = 5'. Using the lemma, we can express '17' as '5q + r', where 'q' is the quotient and 'r' is the remainder. We need to find the values of 'q' and 'r'.

Step 4: Calculation
Dividing '17' by '5', we get 'q = 3' and 'r = 2'. Therefore, '17 = 5(3) + 2', which satisfies the lemma.

Step 5: Importance of Euclid's Division Lemma
Euclid's Division Lemma is a crucial concept in number theory and forms the basis for various mathematical proofs and concepts such as the greatest common divisor (GCD), prime numbers, and modular arithmetic. It helps in solving problems related to divisibility, prime factorization, and finding common factors and multiples.

Conclusion
Euclid's Division Lemma is an essential concept in number theory that provides a framework for understanding division and its properties. It states that any positive integer 'a' can be expressed as the product of a divisor 'b' and a quotient 'q', along with a remainder 'r'. This concept finds wide applications in mathematics and lays the foundation for various other concepts.

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