What are the divisors of 625?

1, 5, 25, 125, 625

There is a total of 5 positive divisors.
The sum of these divisors is 781.
The arithmetic mean is 156.2.

5 odd divisors

1, 5, 25, 125, 625

How to compute the divisors of 625?

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 625 by each of the numbers from 1 to 625 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)

625 / 1 = 625 (the remainder is 0, so 1 is a divisor of 625)
625 / 2 = 312.5 (the remainder is 1, so 2 is not a divisor of 625)
625 / 3 = 208.33333333333 (the remainder is 1, so 3 is not a divisor of 625)
...
625 / 624 = 1.0016025641026 (the remainder is 1, so 624 is not a divisor of 625)
625 / 625 = 1 (the remainder is 0, so 625 is a divisor of 625)

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 625 (i.e. 25). 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 $\frac{N}{d} = D$ )

625 / 1 = 625 (the remainder is 0, so 1 and 625 are divisors of 625)
625 / 2 = 312.5 (the remainder is 1, so 2 is not a divisor of 625)
625 / 3 = 208.33333333333 (the remainder is 1, so 3 is not a divisor of 625)
...
625 / 24 = 26.041666666667 (the remainder is 1, so 24 is not a divisor of 625)
625 / 25 = 25 (the remainder is 0, so 25 and 25 are divisors of 625)