What are the divisors of 691?

1, 691

There is a total of 2 positive divisors.
The sum of these divisors is 692.
The arithmetic mean is 346.

2 odd divisors

1, 691

How to compute the divisors of 691?

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 691 by each of the numbers from 1 to 691 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)

691 / 1 = 691 (the remainder is 0, so 1 is a divisor of 691)
691 / 2 = 345.5 (the remainder is 1, so 2 is not a divisor of 691)
691 / 3 = 230.33333333333 (the remainder is 1, so 3 is not a divisor of 691)
...
691 / 690 = 1.0014492753623 (the remainder is 1, so 690 is not a divisor of 691)
691 / 691 = 1 (the remainder is 0, so 691 is a divisor of 691)

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 691 (i.e. 26.28687885619). 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 $\frac{N}{d} = D$ )

691 / 1 = 691 (the remainder is 0, so 1 and 691 are divisors of 691)
691 / 2 = 345.5 (the remainder is 1, so 2 is not a divisor of 691)
691 / 3 = 230.33333333333 (the remainder is 1, so 3 is not a divisor of 691)
...
691 / 25 = 27.64 (the remainder is 16, so 25 is not a divisor of 691)
691 / 26 = 26.576923076923 (the remainder is 15, so 26 is not a divisor of 691)