What are the divisors of 1825?

1, 5, 25, 73, 365, 1825

There is a total of 6 positive divisors.
The sum of these divisors is 2294.
The arithmetic mean is 382.33333333333.

6 odd divisors

1, 5, 25, 73, 365, 1825

How to compute the divisors of 1825?

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 1825 by each of the numbers from 1 to 1825 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)

1825 / 1 = 1825 (the remainder is 0, so 1 is a divisor of 1825)
1825 / 2 = 912.5 (the remainder is 1, so 2 is not a divisor of 1825)
1825 / 3 = 608.33333333333 (the remainder is 1, so 3 is not a divisor of 1825)
...
1825 / 1824 = 1.000548245614 (the remainder is 1, so 1824 is not a divisor of 1825)
1825 / 1825 = 1 (the remainder is 0, so 1825 is a divisor of 1825)

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 1825 (i.e. 42.720018726588). 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 $\frac{N}{d} = D$ )

1825 / 1 = 1825 (the remainder is 0, so 1 and 1825 are divisors of 1825)
1825 / 2 = 912.5 (the remainder is 1, so 2 is not a divisor of 1825)
1825 / 3 = 608.33333333333 (the remainder is 1, so 3 is not a divisor of 1825)
...
1825 / 41 = 44.512195121951 (the remainder is 21, so 41 is not a divisor of 1825)
1825 / 42 = 43.452380952381 (the remainder is 19, so 42 is not a divisor of 1825)