Euclid's Division Lemma Quiz

Euclid's Division Lemma states that for any two integers a and b, there exists a unique pair of integers q and r such that $a = bq + r$ and $0 \leq r < b$. An example of its application is finding the quotient and remainder when dividing 23 by 5, where a=23 and b=5.

The Euclid Division Algorithm is a method for determining the Highest Common Factor (H.C.F) of two numbers a and b. It involves finding two integers q and r such that $a = bq + r$ and $0 < r < b$ using Euclid’s Division Lemma. If r = 0, the H.C.F is b; otherwise, the algorithm is applied to b and r to generate a new pair of quotients and remainders. This process is repeated until the remainder is zero, at which point the divisor is the H.C.F of the given numbers.

The Fundamental Theorem of Arithmetic states that the prime factorization of a given number is unique. It relates to prime factorization by asserting that regardless of the arrangement of the prime factors, the prime factorization for a given number is always unique.

<p>Prime Factorization refers to the process of representing a natural number as a product of prime numbers. An example is the prime factorization of 36, which is $36 = 2 \times 2 \times 3 \times 3$.</p> Signup and view all the answers

<p>The Least Common Multiple (L.C.M) is significant in finding the smallest positive integer that is a multiple of two or more numbers. It is calculated by finding the prime factorization of each number and then taking the highest power of each prime factor present.</p> Signup and view all the answers

<p>There exists a unique pair of integers $q$ and $r$ such that $a=bq+r$ and $0</p> Signup and view all the answers

<p>It asserts that the prime factorization for a given number is unique, regardless of the arrangement of the prime factors</p> Signup and view all the answers

<p>The H.C.F is $b$</p> Signup and view all the answers

<p>It represents the unique representation of the number as a product of prime numbers</p> Signup and view all the answers

<p>It represents the smallest multiple that is common to both numbers</p> Signup and view all the answers