Divisor

Given two integers d and a, 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. Since 1 and -1 can divide any integer, they are said not to be proper divisors. The number 0 is not considered to be a divisor of any integer.

More examples:

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

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

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

7 divides 7 since $7\cdot 1=7$.

1 divides 5 because $1\cdot 5=5$. It is, however, not a proper divisor.

-3 divides 9 because $(-3)\cdot (-3)=9$

-4 divides -16 because $(-4)\cdot 4=-16$

2 does not divide 9 because there is no integer k such that $2\cdot k=9$. 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 $0\cdot k=0$ 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

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

Related Concepts

(If $d$ is a divisor of $a$ ($d|a$), we say $a$ is a multiple of $d$. For example, since $4|12$, 12 is a multiple of 4. If both $d_{1}$ and $d_{2}<math>aredivisorsof<math>a$, we say $a$ is a common multiple of $d_{1}$ and $d_{2}$. Ignoring the sign (i.e., only considering nonnegative integers), there is a unique greatest common divisor of any two integers $a$ and $b$ written $gcd(a,b)$ or, more commonly, $(a,b)$. 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 $a$ and $b$ is the smallest (positive) integer $m$ such that $a|m$ and $b|m$. Thus, the least common multiple of 12 and 9 is 36 (written $[12,9]=36$).

Further Reading

• 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.