What are the divisors of 1105?

1, 5, 13, 17, 65, 85, 221, 1105

There is a total of 8 positive divisors.
The sum of these divisors is 1512.
The arithmetic mean is 189.

8 odd divisors

1, 5, 13, 17, 65, 85, 221, 1105

How to compute the divisors of 1105?

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 1105 by each of the numbers from 1 to 1105 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)

1105 / 1 = 1105 (the remainder is 0, so 1 is a divisor of 1105)
1105 / 2 = 552.5 (the remainder is 1, so 2 is not a divisor of 1105)
1105 / 3 = 368.33333333333 (the remainder is 1, so 3 is not a divisor of 1105)
...
1105 / 1104 = 1.0009057971014 (the remainder is 1, so 1104 is not a divisor of 1105)
1105 / 1105 = 1 (the remainder is 0, so 1105 is a divisor of 1105)

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 1105 (i.e. 33.241540277189). 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 $\frac{N}{d} = D$ )

1105 / 1 = 1105 (the remainder is 0, so 1 and 1105 are divisors of 1105)
1105 / 2 = 552.5 (the remainder is 1, so 2 is not a divisor of 1105)
1105 / 3 = 368.33333333333 (the remainder is 1, so 3 is not a divisor of 1105)
...
1105 / 32 = 34.53125 (the remainder is 17, so 32 is not a divisor of 1105)
1105 / 33 = 33.484848484848 (the remainder is 16, so 33 is not a divisor of 1105)