What are the divisors of 4025?

1, 5, 7, 23, 25, 35, 115, 161, 175, 575, 805, 4025

There is a total of 12 positive divisors.
The sum of these divisors is 5952.
The arithmetic mean is 496.

12 odd divisors

1, 5, 7, 23, 25, 35, 115, 161, 175, 575, 805, 4025

How to compute the divisors of 4025?

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 4025 by each of the numbers from 1 to 4025 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)

4025 / 1 = 4025 (the remainder is 0, so 1 is a divisor of 4025)
4025 / 2 = 2012.5 (the remainder is 1, so 2 is not a divisor of 4025)
4025 / 3 = 1341.6666666667 (the remainder is 2, so 3 is not a divisor of 4025)
...
4025 / 4024 = 1.0002485089463 (the remainder is 1, so 4024 is not a divisor of 4025)
4025 / 4025 = 1 (the remainder is 0, so 4025 is a divisor of 4025)

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 4025 (i.e. 63.442887702248). 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 $\frac{N}{d} = D$ )

4025 / 1 = 4025 (the remainder is 0, so 1 and 4025 are divisors of 4025)
4025 / 2 = 2012.5 (the remainder is 1, so 2 is not a divisor of 4025)
4025 / 3 = 1341.6666666667 (the remainder is 2, so 3 is not a divisor of 4025)
...
4025 / 62 = 64.91935483871 (the remainder is 57, so 62 is not a divisor of 4025)
4025 / 63 = 63.888888888889 (the remainder is 56, so 63 is not a divisor of 4025)