What are the divisors of 9625?

1, 5, 7, 11, 25, 35, 55, 77, 125, 175, 275, 385, 875, 1375, 1925, 9625

There is a total of 16 positive divisors.
The sum of these divisors is 14976.
The arithmetic mean is 936.

16 odd divisors

1, 5, 7, 11, 25, 35, 55, 77, 125, 175, 275, 385, 875, 1375, 1925, 9625

How to compute the divisors of 9625?

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 9625 by each of the numbers from 1 to 9625 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)

9625 / 1 = 9625 (the remainder is 0, so 1 is a divisor of 9625)
9625 / 2 = 4812.5 (the remainder is 1, so 2 is not a divisor of 9625)
9625 / 3 = 3208.3333333333 (the remainder is 1, so 3 is not a divisor of 9625)
...
9625 / 9624 = 1.0001039068994 (the remainder is 1, so 9624 is not a divisor of 9625)
9625 / 9625 = 1 (the remainder is 0, so 9625 is a divisor of 9625)

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 9625 (i.e. 98.107084351743). 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 $\frac{N}{d} = D$ )

9625 / 1 = 9625 (the remainder is 0, so 1 and 9625 are divisors of 9625)
9625 / 2 = 4812.5 (the remainder is 1, so 2 is not a divisor of 9625)
9625 / 3 = 3208.3333333333 (the remainder is 1, so 3 is not a divisor of 9625)
...
9625 / 97 = 99.226804123711 (the remainder is 22, so 97 is not a divisor of 9625)
9625 / 98 = 98.214285714286 (the remainder is 21, so 98 is not a divisor of 9625)