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A multiple of a number is the product of that number and an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2.
gcd(a, b) is closely related to the least common multiple lcm(a, b): we have gcd(a, b)⋅lcm(a, b) = | a⋅b |. 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. The following versions of distributivity hold true:
lcm(m, n) (least common multiple of m and n) is the product of all prime factors of m or n (with the largest multiplicity for m or n). gcd(m, n) × lcm(m, n) = m × n. Finding the prime factors is often harder than computing gcd and lcm using other algorithms which do not require known prime factorization.
The two integers a and b are coprime if and only if the point with coordinates (a, b) in a Cartesian coordinate system would be "visible" via an unobstructed line of sight from the origin (0, 0), in the sense that there is no point with integer coordinates anywhere on the line segment between the origin and (a, b). (See figure 1.)
The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147. Since ...
LCM may refer to: Computing and mathematics. Latent class model, a concept in statistics; Least common multiple, a function of two integers; Living Computer Museum;
lcm – lowest common multiple (a.k.a. least common multiple) of two numbers. LCHS – locally compact Hausdorff second countable. ld – binary logarithm (log 2). (Also written as lb.) lsc – lower semi-continuity. lerp – linear interpolation. [5] lg – common logarithm (log 10) or binary logarithm (log 2). LHS – left-hand side of an ...
In the case of univariate polynomials, there is a strong relationship between the greatest common divisors and resultants. More precisely, the resultant of two polynomials P , Q is a polynomial function of the coefficients of P and Q which has the value zero if and only if the GCD of P and Q is not constant.