Divisor

Divisor (Number theory)

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.

More examples:

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

5 divides 0 because 5*0 = 0. In fact, every integer except zero divides zero.

7 is a divisor of 49 since 7*7 = 49.

7 divides 7 since 7*1 = 7.

1 divides 5 because 1*5 = 5. In fact, 1 and -1 divide every integer.

-3 divides 9 because (-3)*(-3) = 9.

-4 divides -16 because (-4)*4 = -16.

2 does not divide 9 because there is no integer k such that 2*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 is never a divisor of any number, except of 0 itself (because 0*k=0 for any k, but there is no k such that dk=0 if d is non zero). However, the quotient 0/0 is not defined, as any k would be convenient. Some authors require a divisor to be non zero in the definition in order to avoid this exception.