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Page "Multiplicative function" ¶ 18
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gcd and n
has a solution for x if and only if b is divisible by the greatest common divisor d of a and n ( denoted by gcd ( a, n )).
In particular, there will be exactly d = gcd ( a, n ) solutions in the set of residues
In number theory, Euler's totient or phi function, φ ( n ) is an arithmetic function that counts the number of positive integers less than or equal to n that are relatively prime to n. That is, if n is a positive integer, then φ ( n ) is the number of integers k in the range 1 ≤ kn for which gcd ( n, k ) = 1.
This means that if gcd ( m, n )
0 whenever gcd ( n, k ) > 1.
In particular, division by some v ( mod n ) includes the calculation of the greatest common divisor gcd ( v, n ).
: If the slope is of the form u / v with gcd ( u, n )= 1, then v = 0 ( mod n ) means that the result of the-addition will be, the point at infinity on the curve.
However, if gcd ( v, n ) is neither 1 nor n, then the-addition will not produce a meaningful point on the curve, which shows that our elliptic curve is not a group ( mod n ), but, more importantly for now, gcd ( v, n ) is a non-trivial factor of n.

gcd and k
Proof: Since p is a prime number the only possible values of gcd ( p < sup > k </ sup >, m ) are 1, p, p < sup > 2 </ sup >, ..., p < sup > k </ sup >, and the only way for gcd ( p < sup > k </ sup >, m ) to not equal 1 is for m to be a multiple of p. The multiples of p that are less than or equal to p < sup > k </ sup > are p, 2p, 3p, ..., p < sup > k − 1 </ sup > p = p < sup > k </ sup >, and there are p < sup > k − 1 </ sup > of them.
It is related to their greatest common divisor, gcd ( n, k ), by the formula:
# If gcd ( n, k ) > 1 then χ ( n )
Since gcd ( 1, k )
If gcd ( a, k ) =

gcd and ):
* gcd ( a, b ) is closely related to the least common multiple lcm ( a, b ): we have
Hence gcd ( 455839, 106 )= 1, and working backwards ( a version of the extended Euclidean algorithm ): 1 = 6-5 = 2 · 6-11 = 2 · 28-5 · 11 = 7 · 28-5 · 39 = 7 · 106-19 · 39 = 81707 · 106-19 · 455839.

gcd and greatest
In mathematics, the greatest common divisor ( gcd ), also known as the greatest common factor ( gcf ), or highest common factor ( hcf ), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.
In this article we will denote the greatest common divisor of two integers a and b as gcd ( a, b ).
In a ring all of whose ideals are principal ( a principal ideal domain or PID ), this ideal will be identical with the set of multiples of some ring element d ; then this d is a greatest common divisor of a and b. But the ideal ( a, b ) can be useful even when there is no greatest common divisor of a and b. ( Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.
The original fraction could have also been reduced in a single step by using the greatest common divisor of 90 and 120, which would be gcd ( 90, 120 )= 30.
# For integers, the greatest common divisor is distributive over the least common multiple, and vice versa: gcd ( a, lcm ( b, c ))
* gcd ( m, n ) ( greatest common divisor of m and n ) is the product of all prime factors which are both in m and n ( with the smallest multiplicity for m and n ).
There the contents of a polynomial can be defined as the greatest common divisor of the coefficients of ( like the gcd, the contents is actually a class of associate elements ).
In other words, k is an Erdős – Woods number if there exists a positive integer a such that for each integer i between 0 and k, at least one of the greatest common divisors gcd ( a, a + i ) and gcd ( a + i, a + k ) is greater than 1.

gcd and common
Greatest common divisors can in principle be computed by determining the prime factorizations of the two numbers and comparing factors, as in the following example: to compute gcd ( 18, 84 ), we find the prime factorizations 18 = 2 · 3 < sup > 2 </ sup > and 84 = 2 < sup > 2 </ sup > · 3 · 7 and notice that the " overlap " of the two expressions is 2 · 3 ; so gcd ( 18, 84 ) = 6.
* Every common divisor of a and b is a divisor of gcd ( a, b ).
* If m is a nonzero common divisor of a and b, then gcd ( a / m, b / m ) = gcd ( a, b )/ m.
: This formula is often used to compute least common multiples: one first computes the gcd with Euclid's algorithm and then divides the product of the given numbers by their gcd.
* m and n are coprime ( also called relatively prime ) if gcd ( m, n ) = 1 ( meaning they have no common prime factor ).
2 · gcd ( u / 2, v / 2 ), because 2 is a common divisor.

gcd and divisor
Because gcd ( a, b ) is a divisor of both a and b, it's more efficient to compute the LCM by dividing before multiplying:
Because gcd ( a, b ) is a divisor of both a and b, the division is guaranteed to yield an integer, so the intermediate result can be stored in an integer.
This calculation is always legal and if the gcd of the-coordinate with is unequal to 1 or, so when simplifying fails, then a non-trivial divisor of is found.

gcd and function
* The gcd is a multiplicative function in the following sense: if a < sub > 1 </ sub > and a < sub > 2 </ sub > are relatively prime, then gcd ( a < sub > 1 </ sub >· a < sub > 2 </ sub >, b ) = gcd ( a < sub > 1 </ sub >, b )· gcd ( a < sub > 2 </ sub >, b ).
* The gcd is a commutative function: gcd ( a, b ) = gcd ( b, a ).
* The gcd is an associative function: gcd ( a, gcd ( b, c ))

gcd and where
* gcd ( a, b ), where a and b are not both zero, may be defined alternatively and equivalently as the smallest positive integer d which can be written in the form d = a · p + b · q where p and q are integers.
But it then suffices to go back to the previous gcd term, where, and use the regular Rho algorithm from there.
The condition gcd is equivalent to requiring that the map on is one to one and its inverse is the map where is the multiplicative inverse of.

gcd and is
This representation is convenient for expressions like these for the product, gcd, and lcm:
A much more efficient method is the Euclidean algorithm, which uses a division algorithm such as long division in combination with the observation that the gcd of two numbers also divides their difference.
Then divide 12 by 6 to get a remainder of 0, which means that 6 is the gcd.
* If m is a non-negative integer, then gcd ( m · a, m · b ) = m · gcd ( a, b ).
* If m is any integer, then gcd ( a + m · b, b ) = gcd ( a, b ).

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