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Putting the lcm and gcd into this more general context establishes a duality between them: If a formula involving integer variables, gcd, lcm, ≤ and ≥ is true, then the formula obtained by switching gcd with lcm and switching ≥ with ≤ is also true. (Remember ≤ is defined as divides).
The greatest common divisor (GCD) of integers a and b, at least one of which is nonzero, is the greatest positive integer d such that d is a divisor of both a and b; that is, there are integers e and f such that a = de and b = df, and d is the largest such integer.
Arithmetic billiards is a name given to the process of finding both the LCM and the GCD of two integers using a geometric method. It is named for its similarity to the movement of a billiard ball. [1] To create an arithmetic billiard, a rectangle is drawn with a base of the larger number, and height of the smaller number.
The GCD is said to be the generator of the ideal of a and b. This GCD definition led to the modern abstract algebraic concepts of a principal ideal (an ideal generated by a single element) and a principal ideal domain (a domain in which every ideal is a principal ideal). Certain problems can be solved using this result. [60]
In other words, every GCD domain is a Schreier domain. For every pair of elements x, y of a GCD domain R, a GCD d of x and y and an LCM m of x and y can be chosen such that dm = xy, or stated differently, if x and y are nonzero elements and d is any GCD d of x and y, then xy/d is an LCM of x and y, and vice versa.
As another example, φ(1) = 1 since for n = 1 the only integer in the range from 1 to n is 1 itself, and gcd(1, 1) = 1. Euler's totient function is a multiplicative function , meaning that if two numbers m and n are relatively prime, then φ ( mn ) = φ ( m ) φ ( n ) .
Suppose G is a finite group of order n, and d is a divisor of n.The number of order d elements in G is a multiple of φ(d) (possibly zero), where φ is Euler's totient function, giving the number of positive integers no larger than d and coprime to it.
The only non-trivial fact that this relies on is the relationship between LCM and GCD, which is elementary number theory. I'm going to make the change, please discuss it here if you're opposed. Gaiacarra ( talk ) 21:22, 19 September 2013 (UTC) [ reply ]