What are the divisors of 1025?

1, 5, 25, 41, 205, 1025

There is a total of 6 positive divisors.
The sum of these divisors is 1302.
The arithmetic mean is 217.

6 odd divisors

1, 5, 25, 41, 205, 1025

How to compute the divisors of 1025?

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 1025 by each of the numbers from 1 to 1025 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)

1025 / 1 = 1025 (the remainder is 0, so 1 is a divisor of 1025)
1025 / 2 = 512.5 (the remainder is 1, so 2 is not a divisor of 1025)
1025 / 3 = 341.66666666667 (the remainder is 2, so 3 is not a divisor of 1025)
...
1025 / 1024 = 1.0009765625 (the remainder is 1, so 1024 is not a divisor of 1025)
1025 / 1025 = 1 (the remainder is 0, so 1025 is a divisor of 1025)

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 1025 (i.e. 32.015621187164). 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 $\frac{N}{d} = D$ )

1025 / 1 = 1025 (the remainder is 0, so 1 and 1025 are divisors of 1025)
1025 / 2 = 512.5 (the remainder is 1, so 2 is not a divisor of 1025)
1025 / 3 = 341.66666666667 (the remainder is 2, so 3 is not a divisor of 1025)
...
1025 / 31 = 33.064516129032 (the remainder is 2, so 31 is not a divisor of 1025)
1025 / 32 = 32.03125 (the remainder is 1, so 32 is not a divisor of 1025)