What are the divisors of 1155?

1, 3, 5, 7, 11, 15, 21, 33, 35, 55, 77, 105, 165, 231, 385, 1155

There is a total of 16 positive divisors.
The sum of these divisors is 2304.
The arithmetic mean is 144.

16 odd divisors

1, 3, 5, 7, 11, 15, 21, 33, 35, 55, 77, 105, 165, 231, 385, 1155

How to compute the divisors of 1155?

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 1155 by each of the numbers from 1 to 1155 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)

1155 / 1 = 1155 (the remainder is 0, so 1 is a divisor of 1155)
1155 / 2 = 577.5 (the remainder is 1, so 2 is not a divisor of 1155)
1155 / 3 = 385 (the remainder is 0, so 3 is a divisor of 1155)
...
1155 / 1154 = 1.0008665511265 (the remainder is 1, so 1154 is not a divisor of 1155)
1155 / 1155 = 1 (the remainder is 0, so 1155 is a divisor of 1155)

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 1155 (i.e. 33.985290935933). 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 $\frac{N}{d} = D$ )

1155 / 1 = 1155 (the remainder is 0, so 1 and 1155 are divisors of 1155)
1155 / 2 = 577.5 (the remainder is 1, so 2 is not a divisor of 1155)
1155 / 3 = 385 (the remainder is 0, so 3 and 385 are divisors of 1155)
...
1155 / 32 = 36.09375 (the remainder is 3, so 32 is not a divisor of 1155)
1155 / 33 = 35 (the remainder is 0, so 33 and 35 are divisors of 1155)