What are the divisors of 1955?

1, 5, 17, 23, 85, 115, 391, 1955

There is a total of 8 positive divisors.
The sum of these divisors is 2592.
The arithmetic mean is 324.

8 odd divisors

1, 5, 17, 23, 85, 115, 391, 1955

How to compute the divisors of 1955?

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 1955 by each of the numbers from 1 to 1955 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)

1955 / 1 = 1955 (the remainder is 0, so 1 is a divisor of 1955)
1955 / 2 = 977.5 (the remainder is 1, so 2 is not a divisor of 1955)
1955 / 3 = 651.66666666667 (the remainder is 2, so 3 is not a divisor of 1955)
...
1955 / 1954 = 1.0005117707267 (the remainder is 1, so 1954 is not a divisor of 1955)
1955 / 1955 = 1 (the remainder is 0, so 1955 is a divisor of 1955)

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 1955 (i.e. 44.215381938868). 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 $\frac{N}{d} = D$ )

1955 / 1 = 1955 (the remainder is 0, so 1 and 1955 are divisors of 1955)
1955 / 2 = 977.5 (the remainder is 1, so 2 is not a divisor of 1955)
1955 / 3 = 651.66666666667 (the remainder is 2, so 3 is not a divisor of 1955)
...
1955 / 43 = 45.46511627907 (the remainder is 20, so 43 is not a divisor of 1955)
1955 / 44 = 44.431818181818 (the remainder is 19, so 44 is not a divisor of 1955)