What are the divisors of 967?

1, 967

There is a total of 2 positive divisors.
The sum of these divisors is 968.
The arithmetic mean is 484.

2 odd divisors

1, 967

How to compute the divisors of 967?

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 967 by each of the numbers from 1 to 967 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)

967 / 1 = 967 (the remainder is 0, so 1 is a divisor of 967)
967 / 2 = 483.5 (the remainder is 1, so 2 is not a divisor of 967)
967 / 3 = 322.33333333333 (the remainder is 1, so 3 is not a divisor of 967)
...
967 / 966 = 1.0010351966874 (the remainder is 1, so 966 is not a divisor of 967)
967 / 967 = 1 (the remainder is 0, so 967 is a divisor of 967)

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 967 (i.e. 31.096623610932). 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 $\frac{N}{d} = D$ )

967 / 1 = 967 (the remainder is 0, so 1 and 967 are divisors of 967)
967 / 2 = 483.5 (the remainder is 1, so 2 is not a divisor of 967)
967 / 3 = 322.33333333333 (the remainder is 1, so 3 is not a divisor of 967)
...
967 / 30 = 32.233333333333 (the remainder is 7, so 30 is not a divisor of 967)
967 / 31 = 31.193548387097 (the remainder is 6, so 31 is not a divisor of 967)