What are the divisors of 676?

1, 2, 4, 13, 26, 52, 169, 338, 676

There is a total of 9 positive divisors.
The sum of these divisors is 1281.
The arithmetic mean is 142.33333333333.

6 even divisors

2, 4, 26, 52, 338, 676

3 odd divisors

1, 13, 169

How to compute the divisors of 676?

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 676 by each of the numbers from 1 to 676 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)

676 / 1 = 676 (the remainder is 0, so 1 is a divisor of 676)
676 / 2 = 338 (the remainder is 0, so 2 is a divisor of 676)
676 / 3 = 225.33333333333 (the remainder is 1, so 3 is not a divisor of 676)
...
676 / 675 = 1.0014814814815 (the remainder is 1, so 675 is not a divisor of 676)
676 / 676 = 1 (the remainder is 0, so 676 is a divisor of 676)

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 676 (i.e. 26). 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 $\frac{N}{d} = D$ )

676 / 1 = 676 (the remainder is 0, so 1 and 676 are divisors of 676)
676 / 2 = 338 (the remainder is 0, so 2 and 338 are divisors of 676)
676 / 3 = 225.33333333333 (the remainder is 1, so 3 is not a divisor of 676)
...
676 / 25 = 27.04 (the remainder is 1, so 25 is not a divisor of 676)
676 / 26 = 26 (the remainder is 0, so 26 and 26 are divisors of 676)