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The Art Of Efficiency: 21 Steps To Install Nuclei Scanner – This article is about the greatest common divisor algorithm. For spatial geometry, see Euclidean geometry. For other uses of “Euclid”, see Euclid (disambiguation).

Euclidean’s method for finding the greatest common divisor (GCD) of two initial lengths BA and DC, both defined as a common “unit” shape. Lgth DC is shorter, it is used to “measure” BA, but only once because the rest of EA is less than DC. EA now measures (twice) the shortest length of DC, with the remaining FC shorter than EA. FC measures (three times) the length of EA. Since there is no other, ds system with FC is GCD. On the right is a Nicomachean example with the numbers 49 and 21 generating his GCD of 7 (derived from Heath 1908:300).

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Or Euclid’s algorithm, is an efficient way to calculate the greatest common divisor (GCD) of two number(s), the largest number that divides both without leaving a remainder. It was named after the Greek mathematician Euclid, who first described it in Elemts (

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300 BC). It is an example algorithm, a step-by-step method for performing calculations according to a set of rules, and is one of the oldest and most widely used algorithms. It can be used to reduce fractions to their simplest form and is part of many other calculations and statistical calculations.

The Euclidean algorithm is based on the principle that the largest divisor of two numbers does not change if the larger one and its difference are replaced by a smaller number. For example, 21 is the GCD of 252 and 105 (such as 252 = 21 × 12 and 105 = 21 × 5), and this number 21 is the GCD of 105 and 252 – 105 = 147. Since this substitution reduces the size of . of two numbers, repeating this process results in successively smaller numbers until the two numbers are equal. What happens is that the GCD of two real numbers. By reversing the steps or using the extended Euclidean algorithm, GCD can be expressed as a linear combination of two real numbers, i.e. the sum of two numbers, each multiplied by a number (eg 21 = 5 × 105 + ( – 2 ) × 252). The fact that GCD can always be expressed in this way is called Bezu’s identity.

The Euclidean version of the algorithm described above (also from Euclid) can perform several reduction steps to find a GCD in which one of the given numbers is greater than the other. An improved version of the algorithm skips these steps, instead replacing the two largest numbers with the remainder of the one divided by the second smallest (with this version, the algorithm stops when it reaches a remainder of zero). With this optimization, the algorithm requires no more steps than five times the number of digits (base 10) of the smallest number. Gabriel Lamé proved it in 1844 (Lamé theorem),

And marks the beginning of the theory of complex mathematics. More methods were developed to improve the efficiency of algorithms in the 20th century.

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The Euclidean algorithm has many mathematical and practical applications. It is used for reducing fractions to their simplest form and for division in modern mathematics. Codes using this algorithm have become part of the cryptographic principles used to secure Internet communications, as well as methods of breaking these cryptosystems by generating large numbers. The Euclidean algorithm can be used to solve Diophantine equations, such as finding numbers that satisfy polynomials by the second Chinese theorem, to construct fractions with continued fractions, and to find perfect equations of even numbers that go to real numbers. Finally, it can be used as a regular means of proving number theory theorems such as Lagrange’s quadratic theorem and the isolation of large industries.

The original algorithm was described only for natural numbers and geometric numbers (real numbers), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials of the same variable. This led to modern algebraic concepts such as Euclidean domains.

The Euclidean algorithm calculates the greatest common divisor (GCD) of two natural numbers a and b. The greatest common divisor g is the greatest natural number that divides a and b without leaving a remainder. Synonyms for GCD include greatest common factor (GCF), greatest common factor (HCF), greatest common divisor (HCD), and greatest common factor (GCM). The greatest common divisor is written as gcd(a, b) or, more simply, as (a, b),

Although the latter description is meaningless and is used for concepts such as a target on a number ring, which is related to GCD.

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For example, 6 and 35 are factored as 6=2×3 and 35=5×7, so they are not prime numbers, but their prime factors are different, so 6 and 35 are comparable without any common things other than 1.

A rectangle of size 24 × 60 is covered by a square block t 12 × 12, where 12 is the GCD of 24 and 60. More generally, a rectangle × b can be covered by a square of side c only if c is regular. divisor of a and b.

Let g = gcd (a, b). Since a and b are forms of g, we can write a = mg and b = ng, and there is no greater number G > g for which this is true. The natural numbers m and n must be comparable, since any common factor can be calculated from m and n to increase g. Therefore, any number c that divides a and b must divide g. The greatest common divisor of g a and b is the only common (positive) divisor of a and b such that every common divisor can divide c.

Consider the rectangular area a with b and any divisor c that divides a and b equally. The sides of a rectangle can be divided into segments of length c, which divide the rectangle into a grid of squares of side length c. The GCD of g is the largest value of c for which this is possible. For example, a 24×60 square can be divided into a grid of: 1×1 square, 2×2 square, 3×3 square, 4×4 square, 6×6 square, or 12×12 square. Therefore, 12 is the GCD of 24 and 60. A 24 × 60 square area can be divided into a 12 × 12 square grid, with two squares along one side (24/12 = 2) and five squares along one side (24/ 12 = 2) and five squares along one side (24/12 = 2) along the other (60). /12=5).

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The greatest common divisor of two numbers a and b is the product of the prime factors of the two numbers, where each prime factor can be repeated as many times as it divides a and b.

For example, since 1386 can be calculated as 2 × 3 × 3 × 7 × 11 and 3213 can be calculated as 3 × 3 × 3 × 7 × 17, the GCD of 1386 and 3213 is equal to 63 = 3 × 3 × 7 Product of the most suitable items (with 3 repetitions since 3×3 divides both). If two numbers have no prime factors in common, their GCD is 1 (found here as an example blank pattern); in other words, they are cousins. The main advantage of the Euclidean algorithm is that it can efficiently find the GCD without calculating the principal elements.

Generating large numbers is believed to be a computationally difficult problem, and the security of many widely used theories rests on its impossibility.

The greatest common divisor g of two non-zero numbers a and b is also the smallest positive combination of their linear combinations, that is, the smallest positive number of the form ua + vb where u and v are integers. The set of all linear combinations of a and b is essentially equal to the set of all combinations of g (mg, where m is an integer). In the language of modern mathematics, the ideals a and b correspond best only to g (ideals formed in the same way are called fundamental ideals, and all mathematical theorems are prime ideals). Some properties of GCD are actually easier to see with this statement, for example, the fact that any divisor of a and b also divides GCD (divides the two terms ua + vb). The compatibility of this GCD definition with other definitions is illustrated below.

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The GCD of three or more numbers is equal to the product of the greatest common divisors of all the numbers,

Thus, Euclid’s algorithm, which computes the GCD of two integers, is sufficient to compute the GCD of many arbitrary numbers.

The Euclidean algorithm proceeds in a series of steps, with the output of each step used as input to the next. Track the steps using the integer k, so the first step corresponds to k=0, the next step to k=1, and so on.

R k – 2 = k k r k – 1 + r k with r k

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