Divisor

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Given two integers d and a, where d is nonzero, d is said to divide a, or d is said to be a divisor of a, if and only if there is an integer k such that dk = a. For example, 3 divides 6 because 3 · 2 = 6. Here 3 and 6 play the roles of d and a, while 2 plays the role of k. Though any number divides itself (as does its negative), it is said not to be a proper divisor. The number 0 is not considered to be a divisor of any integer.

More examples:

6 is a divisor of 24 since ${\displaystyle 6\cdot 4=24}$. (We stress that "6 divides 24" and "6 is a divisor of 24" mean the same thing.)

5 divides 0 because ${\displaystyle 5\cdot 0=0}$. In fact, every integer except zero divides zero.

7 is a divisor of 49 since ${\displaystyle 7\cdot 7=49}$.

7 divides 7 since ${\displaystyle 7\cdot 1=7}$.

1 divides 5 because ${\displaystyle 1\cdot 5=5}$.

−3 divides 9 because ${\displaystyle (-3)\cdot (-3)=9}$

−4 divides −16 because ${\displaystyle (-4)\cdot 4=-16}$

2 does not divide 9 because there is no integer k such that ${\displaystyle 2\cdot k=9}$. Since 2 is not a divisor of 9, 9 is said to be an odd integer, or simply an odd number.

• When d is non zero, the number k such that dk = a is unique and is called the exact quotient of a by d, denoted a/d.
• 0 can never be a divisor of any number. It is true that 0 · dk = 0 for any k, however, the quotient 0/0 is not defined, as any k would work. This is the reason 0 is excluded from being considered a divisor.

Notation[edit]

If ${\displaystyle d}$ is a divisor of a (we also say that ${\displaystyle d}$ divides ${\displaystyle a}$, this fact may be expressed by writing ${\displaystyle d|a}$. Similarly, if ${\displaystyle d}$ does not divide ${\displaystyle a}$, we write ${\displaystyle d|a}$. For example, ${\displaystyle 4|12}$ but ${\displaystyle 8|12}$.

Related concepts[edit]

(If ${\displaystyle d}$ is a divisor of ${\displaystyle a}$ (${\displaystyle d|a}$), we say ${\displaystyle a}$ is a multiple of ${\displaystyle d}$. For example, since ${\displaystyle 4|12}$, 12 is a multiple of 4. If both ${\displaystyle d_1}$ and ${\displaystyle d_2}$ are divisors of ${\displaystyle a}$, we say ${\displaystyle a}$ is a common multiple of ${\displaystyle d_1}$ and ${\displaystyle d_2}$. Ignoring the sign (i.e., only considering nonnegative integers), there is a unique greatest common divisor of any two integers ${\displaystyle a}$ and ${\displaystyle b}$ written ${\displaystyle \gcd (a,b)}$ or, more commonly, ${\displaystyle (a,b)}$. The greatest common divisor of 12 and 8 is 4, the greatest common divisor of 15 and 16 is 1. Two numbers with a greatest common divisor of 1 are said to be relatively prime. Complementary to the notion of greatest common divisor is least common multiple. The least common multiple of ${\displaystyle a}$ and ${\displaystyle b}$ is the smallest (positive) integer ${\displaystyle m}$ such that ${\displaystyle a|m}$ and ${\displaystyle b|m}$. Thus, the least common multiple of 12 and 9 is 36 (written ${\displaystyle [12,9]=36}$).

Abstract divisors[edit]

In higher mathematics, the notion of divisor has been abstracted from the integers to the context of general commutative rings. In this setting, they might be termed abstract divisors.

Further reading[edit]

• Scharlau, Winfried; Opolka, Hans (1985). From Fermat to Minkowski: Lectures on the Theory of Numbers and its Historical Development. Springer-Verlag. ISBN 0-387-90942-7.