What are the divisors of 981?

1, 3, 9, 109, 327, 981

There is a total of 6 positive divisors.
The sum of these divisors is 1430.
The arithmetic mean is 238.33333333333.

6 odd divisors

1, 3, 9, 109, 327, 981

How to compute the divisors of 981?

A number N is said to be divisible by a number M (with M non-zero) if, when we divide N by M, the remainder of the division is zero.

$N mod M = 0$

Brute force algorithm

We could start by using a brute-force method which would involve dividing 981 by each of the numbers from 1 to 981 to determine which ones have a remainder equal to 0.

$Remainder = N - (M \times ⌊ \frac{N}{M} ⌋)$

(where $⌊ \frac{N}{M} ⌋$ is the integer part of the quotient)

981 / 1 = 981 (the remainder is 0, so 1 is a divisor of 981)
981 / 2 = 490.5 (the remainder is 1, so 2 is not a divisor of 981)
981 / 3 = 327 (the remainder is 0, so 3 is a divisor of 981)
...
981 / 980 = 1.0010204081633 (the remainder is 1, so 980 is not a divisor of 981)
981 / 981 = 1 (the remainder is 0, so 981 is a divisor of 981)

Improved algorithm using square-root

However, there is another slightly better approach that reduces the number of iterations by testing only integers less than or equal to the square root of 981 (i.e. 31.320919526732). Indeed, if a number N has a divisor D greater than its square root, then there is necessarily a smaller divisor d such that:

$D \times d = N$

(thus, if $\frac{N}{D} = d$ , then $\frac{N}{d} = D$ )

981 / 1 = 981 (the remainder is 0, so 1 and 981 are divisors of 981)
981 / 2 = 490.5 (the remainder is 1, so 2 is not a divisor of 981)
981 / 3 = 327 (the remainder is 0, so 3 and 327 are divisors of 981)
...
981 / 30 = 32.7 (the remainder is 21, so 30 is not a divisor of 981)
981 / 31 = 31.645161290323 (the remainder is 20, so 31 is not a divisor of 981)