Euclid Division Lemma

# Euclid Division Lemma

## Euclid Division Algorithm

The Euclidean division algorithm states that given two positive integers a and b, there is a unique integer q such that a = bq + r, where 0 ≤ r < b. In other words, the algorithm finds the quotient and remainder when a is divided by b.

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The algorithm proceeds as follows:

1. Find the largest integer k that is less than or equal to b and divide a by k to get a quotient q and a remainder r.

2. If r = 0, then q is the desired quotient and a is the desired remainder.

3. If r ≠ 0, then replace a and b with q and r, respectively, and go back to step 1.

## State Euclid’s Division Lemma

The Euclidean division lemma states that for every integer a there exists a unique integer q such that a = q*b + r, where b is the integer closest to a and r is the remainder when a is divided by b.

## Euclid Division Lemma Definition

Given two positive integers, a and b, there exists a unique integer, c, such that a = cb.

## Euclid’s Division Algorithm:

This is an algorithm that is used to divide two numbers. The algorithm is as follows:

If the two numbers are not evenly divisible, then the result will be the quotient and the remainder.

If the two numbers are evenly divisible, then the result will be the whole number that is the result of dividing the two numbers.

## Using Euclid’s Division Algorithm for Finding HCF

Given two positive integers a and b, the highest common factor (HCF) of a and b is the largest positive integer that divides both a and b without leaving a remainder.

The division algorithm is a method for finding the HCF of two numbers. The algorithm is as follows:

1. Divide a by b.

2. If a is not divisible by b, then go to step 1.

3. If a is divisible by b, then the HCF of a and b is b.

The following example illustrates how to use the division algorithm to find the HCF of 24 and 15.

1. Divide 24 by 15.

2. 24 is not divisible by 15, so go to step 1.

3. 15 is divisible by 3, so the HCF of 24 and 15 is 3.

## Euclid’s Lemma:

In a triangle, the sum of the squares of the two shorter sides is equal to the square of the longest side.

The first step is to identify the problem. This may be done through a process of self-evaluation, talking to friends and family, or seeking professional help.

Once the problem is identified, the next step is to develop a plan to address it. This may include making lifestyle changes, seeking counseling or therapy, or taking medication.

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