What are the divisors of 1375?

1, 5, 11, 25, 55, 125, 275, 1375

There is a total of 8 positive divisors.
The sum of these divisors is 1872.
The arithmetic mean is 234.

8 odd divisors

1, 5, 11, 25, 55, 125, 275, 1375

How to compute the divisors of 1375?

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 1375 by each of the numbers from 1 to 1375 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)

1375 / 1 = 1375 (the remainder is 0, so 1 is a divisor of 1375)
1375 / 2 = 687.5 (the remainder is 1, so 2 is not a divisor of 1375)
1375 / 3 = 458.33333333333 (the remainder is 1, so 3 is not a divisor of 1375)
...
1375 / 1374 = 1.0007278020378 (the remainder is 1, so 1374 is not a divisor of 1375)
1375 / 1375 = 1 (the remainder is 0, so 1375 is a divisor of 1375)

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 1375 (i.e. 37.080992435478). 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 $\frac{N}{d} = D$ )

1375 / 1 = 1375 (the remainder is 0, so 1 and 1375 are divisors of 1375)
1375 / 2 = 687.5 (the remainder is 1, so 2 is not a divisor of 1375)
1375 / 3 = 458.33333333333 (the remainder is 1, so 3 is not a divisor of 1375)
...
1375 / 36 = 38.194444444444 (the remainder is 7, so 36 is not a divisor of 1375)
1375 / 37 = 37.162162162162 (the remainder is 6, so 37 is not a divisor of 1375)