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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.
On the right Nicomachus's example with numbers 49 and 21 resulting in their GCD of 7 (derived from Heath 1908:300). In mathematics , the Euclidean algorithm , [ note 1 ] or Euclid's algorithm , is an efficient method for computing the greatest common divisor (GCD) of two integers , the largest number that divides them both without a remainder .
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 may refer to: Computing and mathematics. Latent class model, a concept in statistics; Least common multiple, a function of two integers; Living Computer Museum;
Seventh grade (also 7th Grade or Grade 7) is the seventh year of formal or compulsory education. The seventh grade is typically the first or second year of middle school. In the United States, kids in seventh grade are usually around 12–13 years old. Different terms and numbers are used in other parts of the world.
Leonhard Euler introduced the function in 1763. [7] [8] [9] However, he did not at that time choose any specific symbol to denote it.In a 1784 publication, Euler studied the function further, choosing the Greek letter π to denote it: he wrote πD for "the multitude of numbers less than D, and which have no common divisor with it". [10]
The Motorola 6800 microprocessor was the first for which an undocumented assembly mnemonic HCF became widely known. The operation codes (opcodes—the portions of the machine language instructions that specify an operation to be performed) hexadecimal 9D and DD were reported and given the unofficial mnemonic HCF in a December 1977 article by Gerry Wheeler in BYTE magazine on undocumented ...
The two first subsections, are proofs of the generalized version of Euclid's lemma, namely that: if n divides ab and is coprime with a then it divides b. The original Euclid's lemma follows immediately, since, if n is prime then it divides a or does not divide a in which case it is coprime with a so per the generalized version it divides b.