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The GCD of two numbers a and b is the product of the prime factors shared by the two numbers, where a same prime factor can be used multiple times, but only as long as the product of these factors divides both a and b. For example, since 1386 can be factored into 2 × 3 × 3 × 7 × 11, and 3213 can be factored into 3 × 3 × 3 × 7 × 17, the greatest common divisor of 1386 and 3213 equals 63 = 3 × 3 × 7, the product of their shared prime factors.
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GCD and two
Euclid's method for finding the greatest common divisor ( GCD ) of two starting lengths BA and DC, both defined to be multiples of a common " unit " length.
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor ( GCD ) of two integers, also known as the greatest common factor ( GCF ) or highest common factor ( HCF ).
The GCD of two numbers is the largest number that divides both of them without leaving a remainder.
By reversing the steps in the Euclidean algorithm, the GCD can be expressed as a sum of the two original numbers each multiplied by a positive or negative integer, e. g., 21 = × 105 + × 252.
The Euclidean algorithm calculates the greatest common divisor ( GCD ) of two natural numbers a and b. The greatest common divisor g is the largest natural number that divides both a and b without leaving a remainder.
Thus, Euclid's algorithm, which computes the GCD of two integers, suffices to calculate the GCD of arbitrarily many integers.
* An integral domain is a UFD if and only if it is a GCD domain ( i. e., a domain where every two elements has a greatest common divisor ) satisfying the ascending chain condition on principal ideals.
** GCD domain, an integral domain in which every two non-zero elements have a greatest common divisor
See Cohen, pp. 29 – 31 </ ref > algorithm for calculating the Jacobi symbol, analogous to the Euclidean algorithm for finding the GCD of two numbers.
The following Gödel module is a specification of the greatest common divisor ( GCD ) of two numbers.
The binary GCD algorithm, also known as Stein's algorithm, is an algorithm that computes the greatest common divisor of two nonnegative integers.
To put it algebraically, a sequence of this type is defined by an appropriate choice of two composite numbers a < sub > 1 </ sub > and a < sub > 2 </ sub >, such that the greatest common divisor GCD ( a < sub > 1 </ sub >, a < sub > 2 </ sub >) = 1, and such that for n > 2 there are no primes in the sequence of numbers calculated from the formula
GCD and numbers
On the right Nicomachus ' example with numbers 49 and 21 resulting in their GCD of 7 ( derived from Heath 1908: 300 ).
If implemented using remainders of Euclidean division rather than subtractions, Euclid's algorithm computes the GCD of large numbers efficiently: it never requires more division steps than five times the number of digits ( base 10 ) of the smaller integer.
The GCD of three or more numbers equals the product of the prime factors common to all the numbers, but it can also be calculated by repeatedly taking the GCDs of pairs of numbers.
0 ), the remainders r < sub >− 2 </ sub > and r < sub >− 1 </ sub > equal a and b, the numbers for which the GCD is sought.
There are fast algorithms for computing the GCD that do not require the numbers to be factored, such as the Euclidean algorithm.
This also works for the greatest common divisor ( GCD ), except that instead of multiplying all of the numbers in the Venn diagram, one multiplies only the prime factors that are in the intersection.
But the Euclidean algorithm, using a much more efficient technique, takes only a fraction of a second to compute the GCD for even huge numbers such as these.
However, when the coefficients are integers, rational numbers or polynomials, these arithmetic operations imply a number of GCD computations of coefficients which is of the same order and make the algorithm inefficient.
GCD and b
The greatest common divisor is often written as GCD ( a, b ) or, more simply, as ( a, b ), although the latter notation is also used for other mathematical concepts, such as two-dimensional vectors.
If GCD ( a, b ) = 1, then a and b are said to be coprime ( or relatively prime ).
Some properties of the GCD are in fact easier to see with this description, for instance the fact that any common divisor of a and b also divides the GCD ( it divides both terms of ua + vb ).
: GCD ( a, b, c )
GCD ( GCD ( a, b ), c )
GCD ( b, r < sub > 0 </ sub >)
Compute the GCD of n with the difference ( or sum ) of a and b. This produces a factor, although it may be a trivial factor ( n or 1 ).
DEF GCD ( a, b ) == IF a % b
ELSE IF a-b < b THEN GCD ( b, a-b )
ELSE GCD ( a-b, b )
GCD and is
Because there is no remainder, the process ends with FC being the GCD.
For example, 21 is the GCD of 252 and 105 ( 252 = 12 × 21 ; 105 = 5 × 21 ); since 252 − 105 = ( 12 − 5 ) × 21 = 147, the GCD of 147 and 105 is also 21.
When that occurs, the GCD is the remaining nonzero number.
A 24-by-60 rectangle is covered with ten 12-by-12 square tiles, where 12 is the GCD of 24 and 60.
A key advantage of the Euclidean algorithm is that it can find the GCD efficiently without having to compute the prime factors.
Another definition of the GCD is helpful in advanced mathematics, particularly ring theory.
The equivalence of this GCD definition with the other definitions is described below.
For example, the GCD of 8 and 12 is 4.
A 24-by-60 rectangle is covered with ten 12-by-12 square tiles, where 12 is the GCD of 24 and 60.
The GCD problem is not known to be in NC, and so there is no known way to parallelize its computation across many processors ; nor is it known to be P-complete, which would imply that it is unlikely to be possible to parallelize GCD computation.
GCD and prime
This agrees with the GCD ( 1071, 462 ) found by prime factorization above.
0.091 seconds.