What are the divisors of 971?

1, 971

There is a total of 2 positive divisors.
The sum of these divisors is 972.
The arithmetic mean is 486.

2 odd divisors

1, 971

How to compute the divisors of 971?

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 971 by each of the numbers from 1 to 971 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)

971 / 1 = 971 (the remainder is 0, so 1 is a divisor of 971)
971 / 2 = 485.5 (the remainder is 1, so 2 is not a divisor of 971)
971 / 3 = 323.66666666667 (the remainder is 2, so 3 is not a divisor of 971)
...
971 / 970 = 1.0010309278351 (the remainder is 1, so 970 is not a divisor of 971)
971 / 971 = 1 (the remainder is 0, so 971 is a divisor of 971)

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 971 (i.e. 31.160872901766). 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 $\frac{N}{d} = D$ )

971 / 1 = 971 (the remainder is 0, so 1 and 971 are divisors of 971)
971 / 2 = 485.5 (the remainder is 1, so 2 is not a divisor of 971)
971 / 3 = 323.66666666667 (the remainder is 2, so 3 is not a divisor of 971)
...
971 / 30 = 32.366666666667 (the remainder is 11, so 30 is not a divisor of 971)
971 / 31 = 31.322580645161 (the remainder is 10, so 31 is not a divisor of 971)