Euclid's Division Lemma is a fundamental theorem in number theory and forms the first chapter of Class 10 NCERT Mathematics. It provides the basis for the Euclid's Division Algorithm, which is an efficient method for finding the HCF (Highest Common Factor) of two positive integers.
This lemma is also the basis of RSA encryption, one of the most widely used internet security algorithms. RSA relies on the properties of prime numbers and modular arithmetic, which are deeply connected to Euclid's algorithm.
For any two positive integers 'a' and 'b', there exist unique integers 'q' and 'r' such that:
a = bq + r, where 0 โค r < b
Breaking down the formula:
The condition: The remainder 'r' must always be greater than or equal to 0 AND less than the divisor 'b'.
Step-by-step example: Find the HCF of 455 and 42.
Step 1: Apply lemma: a = bq + r โ 455 = 42 ร 10 + 35 (455 รท 42 = 10 remainder 35)
Step 2: Now take b=42 and r=35 as the new pair: โ 42 = 35 ร 1 + 7 (42 รท 35 = 1 remainder 7)
Step 3: Now take 35 and 7: โ 35 = 7 ร 5 + 0 (35 รท 7 = 5 remainder 0)
When remainder becomes 0, the current divisor is the HCF. โ HCF (455, 42) = 7
The **Lemma** is the mathematical statement/theorem that proves the existence of unique q and r. The **Algorithm** is the step-by-step method (applying the lemma repeatedly) used to actually calculate the HCF.
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