What are the divisors of 1625?

1, 5, 13, 25, 65, 125, 325, 1625

There is a total of 8 positive divisors.
The sum of these divisors is 2184.
The arithmetic mean is 273.

8 odd divisors

1, 5, 13, 25, 65, 125, 325, 1625

How to compute the divisors of 1625?

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 1625 by each of the numbers from 1 to 1625 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)

1625 / 1 = 1625 (the remainder is 0, so 1 is a divisor of 1625)
1625 / 2 = 812.5 (the remainder is 1, so 2 is not a divisor of 1625)
1625 / 3 = 541.66666666667 (the remainder is 2, so 3 is not a divisor of 1625)
...
1625 / 1624 = 1.0006157635468 (the remainder is 1, so 1624 is not a divisor of 1625)
1625 / 1625 = 1 (the remainder is 0, so 1625 is a divisor of 1625)

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 1625 (i.e. 40.311288741493). 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 $\frac{N}{d} = D$ )

1625 / 1 = 1625 (the remainder is 0, so 1 and 1625 are divisors of 1625)
1625 / 2 = 812.5 (the remainder is 1, so 2 is not a divisor of 1625)
1625 / 3 = 541.66666666667 (the remainder is 2, so 3 is not a divisor of 1625)
...
1625 / 39 = 41.666666666667 (the remainder is 26, so 39 is not a divisor of 1625)
1625 / 40 = 40.625 (the remainder is 25, so 40 is not a divisor of 1625)