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 ≤ 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 → 455 = 42 × 10 + 35 (455 ÷ 42 = 10 remainder 35)
Step 2: Now take b=42 and r=35 as the new pair: → 42 = 35 × 1 + 7 (42 ÷ 35 = 1 remainder 7)
Step 3: Now take 35 and 7: → 35 = 7 × 5 + 0 (35 ÷ 7 = 5 remainder 0)
When remainder becomes 0, the current divisor is the HCF. → 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.
Cubes of 1 to 50 — Complete Table and Patterns
Cubes of 1 to 50: complete table from 1³=1 to 50³=125,000. Key values, last digit patterns, cube root tips and solved examples for exams.
Can You Convert Cubic Meter to Square Feet?
Can you convert cubic meter to square feet? Learn why you cannot directly convert a volume unit (3D) into an area unit (2D).
Curved Surface Area (CSA) of a Cone Formula
Learn the formula for the curved surface area of a cone. Understand how to find slant height and solve geometry problems easily.
Curved Surface Area of a Cube
A cube has NO curved surface — its curved surface area (CSA) = 0. Lateral surface area of cube = 4a². Total surface area = 6a². Class 9 Maths explained.
Formula for the Curved Surface Area of a Cylinder (CSA)
Learn the formula for the curved surface area (CSA) of a cylinder (2πrh). Understand the difference between CSA and Total Surface Area (TSA) with examples.
Turn this guide into revision flashcards, a practice exam, or an AI-generated podcast — free, no signup required.