the ability of one number to be divided by another number. The properties of divisibility depend on which aggregates of numbers are examined. If one examines only positive integers, it is said that one number can be divided by another, or, in other words, one is a multiple of the other, if the result (quotient) of dividing the first number (the dividend) by the second (the divisor) is also an integer. A number is said to be a prime if it has no divisors other than itself and 1 (such, for example, are the numbers 2, 3, 5,7, 97, and 199) and composite if otherwise. Any integer can be expressed as the product of prime numbers, for example, 924 = 2.2.3.7.11, and this decomposition is unique to within the order of the factors. For example, the factorization of the number 924 can also be noted in the following manner:
924 = 11.7.3.2.2 = 11.3.2.2.7, and so forth
However, all these factorizations differ only in the order of the factors. The given number n can be divided by the prime number p if and only if p is among the prime factors into which n is decomposed. A number of tests of divisibility have been established, according to which it is easy to determine whether the number n (when written in decimal notation) can be divided by the given prime number o. Of these tests, in practice the following are the most convenient: for divisibility by 2 it is necessary that the last digit be divisible by 2; for divisibility by 3, that the sum of the digits be divisible by 3; for divisibility by 5, that the last digit be either 0 or 5; and for divisibility by 11, that the difference of the sum of the digits in even places and the sum of the digits in odd places be divisible by 11. There are also tests for divisibility by composite numbers: for divisibility by 4 it is necessary that the number represented by the last two digits be divisible by 4; for divisibility by 8, that the number represented by the last three digits be divisible by 8; and for divisibility by 9, that the sum of the digits be divisible by 9. Less convenient are the tests for divisibility by 7 and 13: the difference of the number of thousands and the number expressed by the last three digits should be divisible by these numbers; this operation decreases the number of symbols in the number, and its successive application results in a three-digit number, for example, 825,678 is divisible by 7, since 825 - 678 = 147 is divisible by 7.
For the two numbers a and b, among all their common divisors, there is one that is greatest, which is called the greatest common divisor. If the greatest common divisor of two numbers is equal to 1, then the numbers are called relatively prime. An integer that is divisible by two relatively prime numbers is also divisible by their product. This fact serves as the basis for the simple tests for divisibility by 6 = 2.3, by 10 = 2.5, by 12 = 3.4, by 15 = 3.5, and so forth.
The theory of divisibility of polynomials and algebraic integers is constructed analogously to the theory of divisibility of integers. In the factorization of polynomials, irreducible polynomials play the role of prime numbers. The property of being irreducible depends on which numbers are permitted as coefficients. With real coefficients, polynomials of only the first and second degree can be irreducible, while with complex coefficients, only those of the first degree. The uniqueness will again be arbitrary: with an accuracy of a numerical factor. For algebraic integers the theorem of unique factorization will be incorrect; for example, of the numbers of the type a + b √5 (a and b are integers), the number 4 (for which a = 4, b = 0) permits two factorizations:
4 = 2.2 = (√5 – 1) (√5 – 1)
and none of the factors can be factored further. This fact results in the introduction of the so-called ideal numbers, for which all theorems of factorization hold.