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In mathematics, a divisor of an integer also called a factor of is an integer that may be multiplied by some integer to produce [1] In this case, one also says that is a multiple of An integer is divisible or evenly divisible by another integer if is a divisor of ; this implies dividing by leaves no remainder.
An integer is divisible by a nonzero integer if there exists an integer such that This is written as
This may be read as that divides is a divisor of is a factor of or is a multiple of If does not divide then the notation is [2] [3]
There are two conventions, distinguished by whether is permitted to be zero:
Divisors can be negative as well as positive, although often the term is restricted to positive divisors. For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only the positive ones (1, 2, and 4) would usually be mentioned.
1 and −1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself. Integers divisible by 2 are called even, and integers not divisible by 2 are called odd.
1, −1, and are known as the trivial divisors of A divisor of that is not a trivial divisor is known as a non-trivial divisor (or strict divisor [6] ). A nonzero integer with at least one non-trivial divisor is known as a composite number, while the units −1 and 1 and prime numbers have no non-trivial divisors.
There are divisibility rules that allow one to recognize certain divisors of a number from the number's digits.
There are some elementary rules:
If and then [lower-alpha 2] This is called Euclid's lemma.
If is a prime number and then or
A positive divisor of that is different from is called a proper divisor or an aliquot part of (for example, the proper divisors of 6 are 1, 2, and 3). A number that does not evenly divide but leaves a remainder is sometimes called an aliquant part of
An integer whose only proper divisor is 1 is called a prime number. Equivalently, a prime number is a positive integer that has exactly two positive factors: 1 and itself.
Any positive divisor of is a product of prime divisors of raised to some power. This is a consequence of the fundamental theorem of arithmetic.
A number is said to be perfect if it equals the sum of its proper divisors, deficient if the sum of its proper divisors is less than and abundant if this sum exceeds
The total number of positive divisors of is a multiplicative function meaning that when two numbers and are relatively prime, then For instance, ; the eight divisors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. However, the number of positive divisors is not a totally multiplicative function: if the two numbers and share a common divisor, then it might not be true that The sum of the positive divisors of is another multiplicative function (for example, ). Both of these functions are examples of divisor functions.
If the prime factorization of is given by
then the number of positive divisors of is
and each of the divisors has the form
where for each
For every natural
Also, [7]
where is Euler–Mascheroni constant. One interpretation of this result is that a randomly chosen positive integer n has an average number of divisors of about However, this is a result from the contributions of numbers with "abnormally many" divisors.
In definitions that allow the divisor to be 0, the relation of divisibility turns the set of non-negative integers into a partially ordered set that is a complete distributive lattice. The largest element of this lattice is 0 and the smallest is 1. The meet operation ∧ is given by the greatest common divisor and the join operation ∨ by the least common multiple. This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group Z.
In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n". There is a larger class of number-theoretic functions that do not fit this definition, for example, the prime-counting functions. This article provides links to functions of both classes.
In number theory, two integers a and b are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides a does not divide b, and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also ais prime tob or ais coprime withb.
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest common divisor of x and y is denoted . For example, the GCD of 8 and 12 is 4, that is, gcd(8, 12) = 4.
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm(a, 0) as 0 for all a, since 0 is the only common multiple of a and 0.
In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1) = 1 and whenever a and b are coprime.
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is:
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, 10 = 2 ⋅ 5 is square-free, but 18 = 2 ⋅ 3 ⋅ 3 is not, because 18 is divisible by 9 = 32. The smallest positive square-free numbers are
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as or
In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as or , and may also be called Euler's phi function. In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1. The integers k of this form are sometimes referred to as totatives of n.
In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into the product of non-constant polynomials with rational coefficients.
In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers:
In mathematics, a discrete valuation is an integer valuation on a field K; that is, a function:
In number theory, an n-smooth (or n-friable) number is an integer whose prime factors are all less than or equal to n. For example, a 7-smooth number is a number in which every prime factor is at most 7. Therefore, 49 = 72 and 15750 = 2 × 32 × 53 × 7 are both 7-smooth, while 11 and 702 = 2 × 33 × 13 are not 7-smooth. The term seems to have been coined by Leonard Adleman. Smooth numbers are especially important in cryptography, which relies on factorization of integers. 2-smooth numbers are simply the powers of 2, while 5-smooth numbers are also known as regular numbers.
In number theory, a practical number or panarithmic number is a positive integer such that all smaller positive integers can be represented as sums of distinct divisors of . For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2.
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proven by Euclid in his work Elements. There are several proofs of the theorem.
In mathematical logic, Skolem arithmetic is the first-order theory of the natural numbers with multiplication, named in honor of Thoralf Skolem. The signature of Skolem arithmetic contains only the multiplication operation and equality, omitting the addition operation entirely.
In number theory, the p-adic valuation or p-adic order of an integer n is the exponent of the highest power of the prime number p that divides n. It is denoted . Equivalently, is the exponent to which appears in the prime factorization of .
In mathematics, in the field of algebraic number theory, a modulus is a formal product of places of a global field. It is used to encode ramification data for abelian extensions of a global field.
Anatoly Alexeyevich Karatsuba was a Russian mathematician working in the field of analytic number theory, p-adic numbers and Dirichlet series.
In elementary number theory, the lifting-the-exponent lemma provides several formulas for computing the p-adic valuation of special forms of integers. The lemma is named as such because it describes the steps necessary to "lift" the exponent of in such expressions. It is related to Hensel's lemma.