In mathematics, divisor refers to a positive integer that can divide a given number exactly. For example, the divisors of the number 12 are 1, 2, 3, 4, 6 and 12. Of course, in addition to positive integers, there are negative integers and zeros, but the divisor only includes positive integers. Factors are widely used in number theory, algebra and geometry.

To find the divisor of a number, you can use trial division, that is, divide this number by a positive integer less than or equal to half of this number, and if it is divisible, it is its divisor. In addition, you can also use the prime factor decomposition method, which is to decompose this number into the product of several prime numbers, and then list all possible factors.

In mathematics, there are many formulas related to divisor. Here are some of the more important ones.

1. divisor formula

If a number is decomposed into the product of several prime numbers, the form is p1a1* p2a2 * ... * pn an, where pi is a prime number and ai is a positive integer, and its divisor is (N 1+ 1) * (N2+6538).

For example, if the prime factor of the number 36 is 2 2 * 3 2, then its divisor is (2+ 1) * (2+ 1) = 9, that is, 1, 2,3,4,6,9, 12.

2. Factors and formulas

If you add up all the divisors of a number, the sum is sigma(n). For example, the divisors of the number 12 are 1, 2,3,4,6, 12, and their sum is 1+2+3+4+6+ 12=28.

For any positive integer n, the sum of divisors is sigma (n) = (p1(a1+1)/(p1-1) * (P2 (A2+)

For example, the divisor sum of the number 12 is sigma (12) = (2 (2+1)-1)/(2-1) * (3 (1+65433)

3. Perfect number formula

If the sum of the divisors of a number is equal to itself, then the number is a perfect number. For example, the divisor of the number 6 is 1, 2,3, and their sum is 1+2+3=6, so 6 is a perfect number.

In mathematics, there are some known perfect numbers, such as 6,28,496,8128. However, apart from these known perfect numbers, no other perfect numbers have been found yet.

For a perfect number n, it can be expressed as n = 2 (p- 1) * (2 p- 1), where p is a prime number and 2 p- 1 is also a prime number. This formula is called euler theorem, where 2 p- 1 is called mersenne prime. All the perfect numbers known at present can be expressed by this formula.