What are the divisors of 725?

1, 5, 25, 29, 145, 725

There is a total of 6 positive divisors.
The sum of these divisors is 930.
The arithmetic mean is 155.

6 odd divisors

1, 5, 25, 29, 145, 725

How to compute the divisors of 725?

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 725 by each of the numbers from 1 to 725 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)

725 / 1 = 725 (the remainder is 0, so 1 is a divisor of 725)
725 / 2 = 362.5 (the remainder is 1, so 2 is not a divisor of 725)
725 / 3 = 241.66666666667 (the remainder is 2, so 3 is not a divisor of 725)
...
725 / 724 = 1.0013812154696 (the remainder is 1, so 724 is not a divisor of 725)
725 / 725 = 1 (the remainder is 0, so 725 is a divisor of 725)

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 725 (i.e. 26.925824035673). 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 $\frac{N}{d} = D$ )

725 / 1 = 725 (the remainder is 0, so 1 and 725 are divisors of 725)
725 / 2 = 362.5 (the remainder is 1, so 2 is not a divisor of 725)
725 / 3 = 241.66666666667 (the remainder is 2, so 3 is not a divisor of 725)
...
725 / 25 = 29 (the remainder is 0, so 25 and 29 are divisors of 725)
725 / 26 = 27.884615384615 (the remainder is 23, so 26 is not a divisor of 725)