What are the divisors of 961?

1, 31, 961

There is a total of 3 positive divisors.
The sum of these divisors is 993.
The arithmetic mean is 331.

3 odd divisors

1, 31, 961

How to compute the divisors of 961?

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 961 by each of the numbers from 1 to 961 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)

961 / 1 = 961 (the remainder is 0, so 1 is a divisor of 961)
961 / 2 = 480.5 (the remainder is 1, so 2 is not a divisor of 961)
961 / 3 = 320.33333333333 (the remainder is 1, so 3 is not a divisor of 961)
...
961 / 960 = 1.0010416666667 (the remainder is 1, so 960 is not a divisor of 961)
961 / 961 = 1 (the remainder is 0, so 961 is a divisor of 961)

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 961 (i.e. 31). 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 $\frac{N}{d} = D$ )

961 / 1 = 961 (the remainder is 0, so 1 and 961 are divisors of 961)
961 / 2 = 480.5 (the remainder is 1, so 2 is not a divisor of 961)
961 / 3 = 320.33333333333 (the remainder is 1, so 3 is not a divisor of 961)
...
961 / 30 = 32.033333333333 (the remainder is 1, so 30 is not a divisor of 961)
961 / 31 = 31 (the remainder is 0, so 31 and 31 are divisors of 961)