Euclid's Division Lemma Quiz
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Questions and Answers

What is an algorithm?

A series of well-defined steps which gives a procedure for solving a type of problem.

What is a lemma?

A proven statement used for proving another statement.

Euclid's division algorithm finds the Highest Common Factor (HCF) of two given positive integers.

True

What is the condition for the remainder in Euclid's division lemma?

<p>The remainder is less than the divisor.</p> Signup and view all the answers

In the equation a = bq + r, the integers q and r are uniquely determined for any given integers a and b.

<p>True</p> Signup and view all the answers

What are the integers q and r called in Euclid's division lemma?

<p>Quotient and remainder</p> Signup and view all the answers

Study Notes

Euclid's Division Lemma

  • Euclid's division lemma is a technique used to compute the highest common factor (HCF) of two given positive integers
  • Given positive integers a and b, with a > b, there exist unique integers q and r, such that a = bq + r, where 0 ≤ r < b
  • The integers q and r are called the quotient and remainder, respectively
  • This result has been known for a long time, but Euclid's division algorithm was first recorded in Book VII of Euclid's Elements
  • The steps involved in applying Euclid's division lemma are as follows:
    • Apply Euclid's division lemma to c and d such that c > d.
    • If r = 0, d is the HCF of c and d.
    • Continue the process until the remainder is zero. The divisor at this stage will be the required HCF.

Example

  • A trader was selling eggs and had a dispute with the Panchayat
  • The trader was asked how many eggs were there in total given the following conditions:
    • The number of eggs, when counted in pairs, leaves a remainder of 1.
    • The number of eggs, when counted in threes, leaves a remainder of 2.
    • The number of eggs, when counted in fours, leaves a remainder of 3.
    • The number of eggs, when counted in fives, leaves a remainder of 4.
    • The number of eggs, when counted in sevens, leaves no remainder.
  • The number of eggs must be less than 150.
  • The solution would be found by working backwards to solve for the number of eggs

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Description

Test your knowledge of Euclid's Division Lemma and its application in computing the highest common factor (HCF) of two integers. This quiz covers key concepts, steps involved in the algorithm, and practical examples. Perfect for students learning number theory.

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