*For the second operand of a division, see division (mathematics).*
In mathematics, a **divisor** of an integer *n*, also called a **factor** of *n*, is an integer which divides *n* without leaving a remainder. The statement "*m* is a divisor of *n*" can be written as {\displaystyle m\mid n}. Any number is always divisible by 1 and itself, which are two of the divisors. A prime number is a number with no other divisors. The positive divisors of a number *n*, other than *n* itself, are the **proper divisors** of *n*.

Finding one or more factors of a given number is called factorization.

Explanation
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For example, 7 is a divisor of 42 because 42÷7 = 6. We also say that "42 is **divisible** by 7", "42 is a **multiple** of 7", "7 **divides** 42", or "7 is a **factor** of 42", and we usually write 7 | 42. For example, the positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.

In general, we say that *m* divides *n* for non-zero integers *m* and *n*, if and only if there exists an integer *k* such that *n* = *km*. Thus, divisors can be negative as well as positive, although we often restrict our attention to positive divisors. (For example, there are six divisors of four, 1, 2, 4, -1, -2, -4, but one would usually mention only the positive ones, 1, 2, and 4.) 

1 and -1 divide (are divisors of) every integer, every integer 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.

A divisor of *n* that is not 1, -1, *n* or -*n* is known as a **non-trivial divisor**; numbers with non-trivial divisors are known as composite numbers, while prime numbers have no non-trivial divisors.

The name comes from the arithmetic operation of division: if *a÷b* = *c*, then *a* is the dividend, *b* the **divisor,** and *c* the quotient.

Spotting divisors
-----------------

There are properties which allow one to recognize certain divisors of a number from the number's digits. Those properties can be used as "math tricks" to quickly spot some divisors of a number.

For example, if the last digit is even (0, 2, 4, 6 or 8), then 2 is a divisor. If the last digit is 0 or 5, then 5 is a divisor. If the digits add up to a multiple of 3, then 3 is a divisor. For the number 340, ending in "0" then both 2 and 5 are divisors, plus 2×5 = 10 is also a divisor. Dividing by 10, 340/10 = 34, as finally 2×17. Combining all the smaller numbers, the 12 divisors of 340 are:

* Divisors of 340: 1, 2, 4, 5, 10, 17, 20, 34, 68, 85, 170, 340.

Note that any number is always evenly divisible by 1 and itself.

Related pages
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* Coprime
* Euclid's lemma

Other websites
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* Online Number Factorizer Instantly factors numbers up to 17 digits long
* Factoring Calculator Archived 2005-04-04 at the Wayback Machine -- Factoring calculator that displays the prime factors and the prime and non-prime divisors of a given number.



| * v
* t
* e
Divisibility-based sets of integers |
| --- |
| Overview | 
* Integer factorization
* Divisor
* Unitary divisor
* Divisor function
* Prime factor
* Fundamental theorem of arithmetic

 | Divisibility of 60 |
| Factorization forms | 
* Prime
* Composite
* Semiprime
* Pronic
* Sphenic
* Square-free
* Powerful
* Perfect power
* Achilles
* Smooth
* Regular
* Rough
* Unusual

 |
| Constrained divisor sums | 
* Perfect
* Almost perfect
* Quasiperfect
* Multiply perfect
* Hemiperfect
* Hyperperfect
* Superperfect
* Unitary perfect
* Semiperfect
* Practical
* Erdős–Nicolas

 |
| With many divisors | 
* Abundant
* Primitive abundant
* Highly abundant
* Superabundant
* Colossally abundant
* Highly composite
* Superior highly composite
* Weird

 |
| Aliquot sequence-related | 
* Untouchable
* Amicable (Triple)
* Sociable
* Betrothed

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| Base-dependent | 
* Equidigital
* Extravagant
* Frugal
* Harshad
* Polydivisible
* Smith

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| Other sets | 
* Arithmetic
* Deficient
* Friendly
* Solitary
* Sublime
* Harmonic divisor
* Descartes
* Refactorable
* Superperfect

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