What are the divisors of 325?

1, 5, 13, 25, 65, 325

There is a total of 6 positive divisors.
The sum of these divisors is 434.
The arithmetic mean is 72.333333333333.

6 odd divisors

1, 5, 13, 25, 65, 325

How to compute the divisors of 325?

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 325 by each of the numbers from 1 to 325 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)

325 / 1 = 325 (the remainder is 0, so 1 is a divisor of 325)
325 / 2 = 162.5 (the remainder is 1, so 2 is not a divisor of 325)
325 / 3 = 108.33333333333 (the remainder is 1, so 3 is not a divisor of 325)
...
325 / 324 = 1.0030864197531 (the remainder is 1, so 324 is not a divisor of 325)
325 / 325 = 1 (the remainder is 0, so 325 is a divisor of 325)

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 325 (i.e. 18.02775637732). 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 $\frac{N}{d} = D$ )

325 / 1 = 325 (the remainder is 0, so 1 and 325 are divisors of 325)
325 / 2 = 162.5 (the remainder is 1, so 2 is not a divisor of 325)
325 / 3 = 108.33333333333 (the remainder is 1, so 3 is not a divisor of 325)
...
325 / 17 = 19.117647058824 (the remainder is 2, so 17 is not a divisor of 325)
325 / 18 = 18.055555555556 (the remainder is 1, so 18 is not a divisor of 325)