Divisor

Divisor (Number theory)

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.

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.

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.

• Note that 0 is never a divisor of any number. For example, if 0 were to divide 8, there would have to be an integer k such that 0*k = 8, which is impossible. (Nor does 0 divide 0, by convention rather than impossibility.)