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 ${\displaystyle m\mid n}$.^[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.

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.

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.

• Coprime
• Euclid's lemma

1. ↑ "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-09-20.
2. ↑ Weisstein, Eric W. "Divisor". mathworld.wolfram.com. Retrieved 2020-09-20.
3. ↑ Caldwell, Chris. "Divisor". Prime Glossary. Retrieved September 20, 2020.

• 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.