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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.
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.
14, 49, −21 and 0 are multiples of 7, whereas 3 and −6 are not. This is because there are integers that 7 may be multiplied by to reach the values of 14, 49, 0 and −21, while there are no such integers for 3 and −6.
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:
If the product of two characters is defined by pointwise multiplication ... [50] in 1956, and proved in 2017 by Klurman and Mangerel. [51] See also.
The index of 1 / k where n / i+1 < k ≤ n / i and n is the least common multiple of the first i numbers, n = lcm([2, i]), is given by: [8] (/) = + = (). A similar expression was used as an approximation of I n ( x ) {\displaystyle I_{n}(x)} for low values of x {\displaystyle x} in the classical paper by F. Dress [ 9 ] .
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 some number, and any prime number measure the product, it will also measure one of the original numbers.
In 493 AD, Victorius of Aquitaine wrote a 98-column multiplication table which gave (in Roman numerals) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then the fractions down to 1/144." [6]
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