What are the divisors of 991?

1, 991

There is a total of 2 positive divisors.
The sum of these divisors is 992.
The arithmetic mean is 496.

2 odd divisors

1, 991

How to compute the divisors of 991?

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 991 by each of the numbers from 1 to 991 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)

991 / 1 = 991 (the remainder is 0, so 1 is a divisor of 991)
991 / 2 = 495.5 (the remainder is 1, so 2 is not a divisor of 991)
991 / 3 = 330.33333333333 (the remainder is 1, so 3 is not a divisor of 991)
...
991 / 990 = 1.0010101010101 (the remainder is 1, so 990 is not a divisor of 991)
991 / 991 = 1 (the remainder is 0, so 991 is a divisor of 991)

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 991 (i.e. 31.480152477394). 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 $\frac{N}{d} = D$ )

991 / 1 = 991 (the remainder is 0, so 1 and 991 are divisors of 991)
991 / 2 = 495.5 (the remainder is 1, so 2 is not a divisor of 991)
991 / 3 = 330.33333333333 (the remainder is 1, so 3 is not a divisor of 991)
...
991 / 30 = 33.033333333333 (the remainder is 1, so 30 is not a divisor of 991)
991 / 31 = 31.967741935484 (the remainder is 30, so 31 is not a divisor of 991)