Divisors

Compute Divisors and Number of Divisors

Studying divisibility of integers by other integers is a common task in number theory. The MuPAD® numlib library contains the functions that support this task. These functions return all divisors, the sum of all divisors, the number of divisors, and the number of prime divisors. For example, to find all positive integer divisors of an integer number, use the numlib::divisors function:

numlib::divisors(12345)

To find only prime divisors of an integer, use the numlib::primedivisors function:

numlib::primedivisors(12345)

To compute the number of all divisors of an integer, use the numlib::numdivisors function. To compute the number of prime divisors, use the numlib::numprimedivisors function. For example, the number 123456789987654321 has 192 divisors. Only seven of these divisors are prime numbers:

numlib::numdivisors(123456789987654321),
numlib::numprimedivisors(123456789987654321)

The numlib::numprimedivisors function does not take into account multiplicities of prime divisors. This function counts a prime divisor with multiplicity as one prime divisor. To compute the sum of multiplicities of prime divisors, use the numlib::Omega function. For example, the number 27648 has 44 divisors, and 2 of them are prime numbers. The prime divisors of 27648 have multiplicities; the total sum of these multiplicities is 13:

numlib::numdivisors(27648),
numlib::numprimedivisors(27648),
numlib::Omega(27648)

You can factor the number 27648 into prime numbers to reveal the multiplicities. To factor an integer into primes, use the ifactor function:

ifactor(27648)

To compute the sum of all positive integer divisors of an integer number, use the numlib::sumdivisors function. For example, compute the sum of positive divisors of the number 12345:

numlib::sumdivisors(12345)

Compute Greatest Common Divisors

The largest nonnegative integer that divides all the integers of a sequence exactly (without remainders) is called the greatest common divisor of a sequence. To compute the greatest common divisor of a sequence of integers, use the igcd function. For example, compute the greatest common divisor of the following numbers:

igcd(12345, 23451, 34512, 45123, 51234)

The icontent function computes the greatest common divisor of the coefficients of a polynomial. All coefficients must be integers:

icontent(12*x^2 + 16*x + 24)

Compute Least Common Multiples

The smallest integer that is exactly divisible (without remainders) by all integers of a sequence is called the least common multiple of a sequence. To compute the least common multiple of a sequence of integers, use the ilcm function. For example, compute the least common multiple of the following numbers:

ilcm(12, 5, 2, 21)

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