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In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm (a, b), is the smallest positive integer that is divisible by both a and b. [1][2] Since division of integers by zero is undefined, this definition has meaning only if a and b are both ...
The least common multiple of a and b is equal to their product ab, i.e. lcm ... the integers 6, 10, 15 are coprime because 1 is the only positive integer that divides ...
Appearance. In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest common divisor of x and y is denoted . For example, the GCD of 8 and 12 is 4, that is ...
The lowest common denominator of a set of fractions is the lowest number that is a multiple of all the denominators: their lowest common multiple. The product of the denominators is always a common denominator, as in: but it is not always the lowest common denominator, as in: Here, 36 is the least common multiple of 12 and 18.
The greatest common divisor g of a and b is the unique (positive) common divisor of a and b that is divisible by any other common divisor c. [6] The greatest common divisor can be visualized as follows. [7] Consider a rectangular area a by b, and any common divisor c that divides both a and b exactly.
The fundamental theorem can be derived from Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of Euclid's Elements. If two numbers by multiplying one another make somenumber, and any prime number measure the product, it willalso measure one of the original numbers. — Euclid, Elements Book VII, Proposition 30.
Equivalently, g(n) is the largest least common multiple (lcm) of any partition of n, or the maximum number of times a permutation of n elements can be recursively applied to itself before it returns to its starting sequence. For instance, 5 = 2 + 3 and lcm(2,3) = 6. No other partition of 5 yields a bigger lcm, so g(5) = 6.
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.