|addend + addend =||sum|
|minuend − subtrahend =||difference|
|multiplicand × multiplier =||product|
|dividend ÷ divisor =||quotient|
|nth root (√)|
|degree √ =||root|
In general, for non-zero integers and , it is said that divides —and, dually, that is divisible by —written:
if there exists an integer such that . Thus, divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. (For example, there are six divisors of four, 1, 2, 4, −1, −2, −4, but only the positive ones would usually be mentioned, i.e. 1, 2, and 4.)
1 and −1 divide (are divisors of) every integer, every integer (and its negation) is a divisor of itself, and every integer is a divisor of 0, except by convention 0 itself (see also division by zero). Numbers divisible by 2 are called even and numbers not divisible by 2 are called odd.
1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a trivial divisor is known as a non-trivial divisor. A number 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 which allow one to recognize certain divisors of a number from the number's digits.
- 7 is a divisor of 42 because , so we can say . It can also be said that 42 is divisible by 7, 42 is a multiple of 7, 7 divides 42, or 7 is a factor of 42.
- The non-trivial divisors of 6 are 2, −2, 3, −3.
- The positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.
- The set of all positive divisors of 60, , partially ordered by divisibility, has the Hasse diagram:
Further notions and facts 
There are some elementary rules:
- If and , then . This is the transitive relation.
- If and , then or .
- If and , then it is NOT always true that (e.g. and but 5 does not divide 6). However, when and , then is true, as is .
If is a prime number and then or .
A positive divisor of which is different from is called a proper divisor or an aliquot part of . A number that does not evenly divide but leaves a remainder is 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 which has exactly two positive factors: 1 and itself.
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 (e.g. ). 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 , .
where is Euler–Mascheroni constant. One interpretation of this result is that a randomly chosen positive integer n has an expected number of divisors of about .
In abstract algebra 
The relation of divisibility turns the set of non-negative integers into a partially ordered set, in fact into 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 v by the least common multiple. This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group .
See also 
- Arithmetic functions
- Divisibility rule
- Divisor function
- Euclid's algorithm
- Fraction (mathematics)
- Table of divisors — A table of prime and non-prime divisors for 1–1000
- Table of prime factors — A table of prime factors for 1–1000
- Durbin, John R. (1992). Modern Algebra : an Introduction (3rd ed. ed.). New York: Wiley. p. 61. ISBN 0-471-51001-7. "An integer is divisible by an integer if there is an integer (for quotient) such that ."
- Hardy, G. H.; E. M. Wright (April 17, 1980). An Introduction to the Theory of Numbers. Oxford University Press. p. 264. ISBN 0-19-853171-0.
- Richard K. Guy, Unsolved Problems in Number Theory (3rd ed), Springer Verlag, 2004 ISBN 0-387-20860-7; section B.
- Øystein Ore, Number Theory and its History, McGraw–Hill, NY, 1944 (and Dover reprints).