Euclid's Division Lemma:
Euclid's Division Lemma states that for any positive integers a and b, there exist unique non-negative integers q and r such that a = b × q + r, where r < />

Explanation:
Euclid's Division Lemma is a fundamental concept in number theory that helps in understanding the properties and relationships between numbers. It is used to divide a given number (the dividend) by another number (the divisor) and obtain the quotient and remainder.

Let's take an example to understand Euclid's Division Lemma in detail using the numbers 900 and 270.

Step 1: Divisor (b)
The divisor in this case is 270. It is the number by which we divide the dividend (900).

Step 2: Quotient (q)
The quotient is the number of times the divisor can be completely divided by the dividend. It represents the whole number part of the division.

To find the quotient, we divide the dividend (900) by the divisor (270):
900 ÷ 270 = 3

Therefore, the quotient (q) is 3.

Step 3: Remainder (r)
The remainder is the amount left after dividing the dividend by the divisor. It represents the fractional or decimal part of the division.

To find the remainder, we use the formula: remainder = dividend - (divisor × quotient)
remainder = 900 - (270 × 3)
remainder = 900 - 810
remainder = 90

Therefore, the remainder (r) is 90.

Step 4: Applying Euclid's Division Lemma
According to Euclid's Division Lemma, we can write the original number (900) as the sum of the product of the divisor (270) and the quotient (3), and the remainder (90):
900 = 270 × 3 + 90

This equation satisfies Euclid's Division Lemma as the remainder (90) is less than the divisor (270).

Conclusion:
In conclusion, Euclid's Division Lemma is a powerful concept that helps in understanding the division of numbers and finding their quotient and remainder. It is a fundamental tool in number theory and is widely used in various mathematical applications.