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# Divisor

*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 [math]m \mid n[/math].^{[1]}^{[2]} 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*.^{[3]}

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

## Explanation

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

## References

- ↑ "Compendium of Mathematical Symbols" (in en-US). 2020-03-01. https://mathvault.ca/hub/higher-math/math-symbols/.
- ↑ Weisstein, Eric W.. "Divisor" (in en). https://mathworld.wolfram.com/Divisor.html.
- ↑ Caldwell, Chris. "Divisor". https://primes.utm.edu/glossary/page.php?sort=Divisor.

## Other websites

- Online Number Factorizer Instantly factors numbers up to 17 digits long
- Factoring Calculator -- Factoring calculator that displays the prime factors and the prime and non-prime divisors of a given number.