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The GCD is a multiplicative function in the following sense: if a 1 and a 2 are relatively prime, then gcd(a 1 ⋅a 2, b) = gcd(a 1, b)⋅gcd(a 2, b). gcd( a , b ) is closely related to the least common multiple lcm( a , b ) : we have
Since b ≥ φ N−1, then N − 1 ≤ log φ b. Since log 10 φ > 1/5, (N − 1)/5 < log 10 φ log φ b = log 10 b. Thus, N ≤ 5 log 10 b. Thus, the Euclidean algorithm always needs less than O divisions, where h is the number of digits in the smaller number b.
The study of GCD type matrices originates from Smith (1875) who evaluated the determinant of certain GCD and LCM matrices. Smith showed among others that the determinant of the n × n {\displaystyle n\times n} matrix ( gcd ( i , j ) ) {\displaystyle (\gcd(i,j))} is ϕ ( 1 ) ϕ ( 2 ) ⋯ ϕ ( n ) {\displaystyle \phi (1)\phi (2)\cdots \phi (n ...
The assumption is that the loop must be normalized – written so that the loop index/variable starts at 1 and gets incremented by 1 in every iteration. For example, in the following loop, a=2, b=3, c=2, d=0 and GCD(a,c)=2 and (d-b) is -3. Since 2 does not divide -3, no dependence is possible.
For example, if you had two types of coins valued at 6 cents and 14 cents, the GCD would equal 2, and there would be no way to combine any number of such coins to produce a sum which was an odd number; additionally, even numbers 2, 4, 8, 10, 16 and 22 (less than m=24) could not be formed, either.
An example of the latter is a(x) = x+1, b(x) = x−1 with ab(x) = x. If ab = ba , we can at least say that ord( ab ) divides lcm (ord( a ), ord( b )). As a consequence, one can prove that in a finite abelian group, if m denotes the maximum of all the orders of the group's elements, then every element's order divides m .
In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1. [2] [3] The integers k of this form are sometimes referred to as totatives of n. For example, the totatives of n = 9 are the six numbers 1, 2, 4, 5, 7 and 8.
There are infinitely many pseudoprimes to any given base a > 1. In 1904, Cipolla showed how to produce an infinite number of pseudoprimes to base a > 1: let A = (a p - 1)/(a - 1) and let B = (a p + 1)/(a + 1), where p is a prime number that does not divide a(a 2 - 1).
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