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 (A)

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 (A)</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

Flashcards

Euclid's Division Lemma

Given positive integers 'a' and 'b', there exist unique integers 'q' (quotient) and 'r' (remainder) satisfying a = bq + r, where 0 ≤ r < b.

Lemma

A proven statement used to prove another statement.

Algorithm

A series of well-defined steps that solve a specific type of problem.

Highest Common Factor (HCF)

The largest positive integer that divides both 'a' and 'b'.

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Euclid's Division Algorithm

Technique to find the HCF of two positive integers.

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Fundamental Theorem of Arithmetic

Every composite number can be expressed uniquely as a product of prime numbers.

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Prime Factorization

Process of expressing a composite number as a product of its prime factors.

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Lowest Common Multiple (LCM)

The smallest positive integer divisible by both 'a' and 'b'.

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Prime Number

A number that cannot be expressed as a product of two smaller natural numbers, excluding 1.

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Composite Number

A natural number that is not prime.

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Rational Number

A number that can be expressed as a fraction p/q, where p and q are integers and q is not zero.

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Irrational Number

A number that cannot be expressed as a simple fraction.

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Theorem 1.3

If 'p' divides 'a²', then 'p' divides 'a', where 'p' is a prime number and 'a' is a positive integer.

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Theorem 1.4

√2 is irrational.

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Terminating or Repeating Decimal

A number that can be written as a decimal that either terminates or repeats.

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Non-terminating, Non-repeating Decimal

A decimal that goes on forever without repeating.

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Prime Factorization and Decimal Expansion

The decimal expansion of a rational number p/q (where q ≠ 0) can be determined by the prime factorization of the denominator 'q'.

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Real Number

A number that represents a point on a number line, including integers, fractions, decimals, and irrational numbers.

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Division Operation

To divide a smaller number 'a' by a larger number 'b' to find the unique quotient 'q' and remainder 'r'.

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Even Number

Any number that can be divided by 2.

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Odd Number

Any number that cannot be divided by 2.

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Integer

A value representing a specific point on a number line.

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Function

A set of ordered pairs that represent a relationship between two variables.

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Exponents

The expression representing the number of times the base is multiplied by itself.

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Base Number

The number on which the exponent acts.

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Addition

The fundamental mathematical operation of combining two numbers to get a new number.

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Subtraction

The fundamental mathematical operation of taking away one quantity from another.

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Multiplication

The fundamental mathematical operation of finding the total number of units in multiple identical groups.

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Division

The fundamental mathematical operation of dividing a number by another.

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Negative Number

The opposite of the original number, with the same magnitude.

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Natural Numbers

The set of positive whole numbers, excluding zero.

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Integers

The set of positive and negative whole numbers, including zero.

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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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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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