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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.
m and n are coprime (also called relatively prime) if gcd(m, n) = 1 (meaning they have no common prime factor). 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 ...
The natural numbers m and n must be coprime, since any common factor could be factored out of m and n to make g greater. Thus, any other number c that divides both a and b must also divide g. 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]
a highly abundant number has a sum of positive divisors that is greater than any lesser number; that is, σ(n) ... 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 320, 640 ...
[39] [40] The factorial number system is a mixed radix notation for numbers in which the place values of each digit are factorials. [ 41 ] Factorials are used extensively in probability theory , for instance in the Poisson distribution [ 42 ] and in the probabilities of random permutations . [ 43 ]
This is equivalent to their greatest common divisor (GCD) being 1. [2] One says also a is prime to b or a is coprime with b. The numbers 8 and 9 are coprime, despite the fact that neither—considered individually—is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both ...
For example, if n = 171 × p × q where p < q are very large primes, trial division will quickly produce the factors 3 and 19 but will take p divisions to find the next factor. As a contrasting example, if n is the product of the primes 13729, 1372933, and 18848997161, where 13729 × 1372933 = 18848997157, Fermat's factorization method will ...
This is useful in computer science, since most data structures have members with 2 X possible values. For example, a byte has 256 (2 8) possible values (0–255). Therefore, to fill a byte or bytes with random values, a random number generator that produces values 1–256 can be used, the byte taking the output value −1.