What are the divisors of 1085?

1, 5, 7, 31, 35, 155, 217, 1085

There is a total of 8 positive divisors.
The sum of these divisors is 1536.
The arithmetic mean is 192.

8 odd divisors

1, 5, 7, 31, 35, 155, 217, 1085

How to compute the divisors of 1085?

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 1085 by each of the numbers from 1 to 1085 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)

1085 / 1 = 1085 (the remainder is 0, so 1 is a divisor of 1085)
1085 / 2 = 542.5 (the remainder is 1, so 2 is not a divisor of 1085)
1085 / 3 = 361.66666666667 (the remainder is 2, so 3 is not a divisor of 1085)
...
1085 / 1084 = 1.0009225092251 (the remainder is 1, so 1084 is not a divisor of 1085)
1085 / 1085 = 1 (the remainder is 0, so 1085 is a divisor of 1085)

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 1085 (i.e. 32.939338184001). 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 $\frac{N}{d} = D$ )

1085 / 1 = 1085 (the remainder is 0, so 1 and 1085 are divisors of 1085)
1085 / 2 = 542.5 (the remainder is 1, so 2 is not a divisor of 1085)
1085 / 3 = 361.66666666667 (the remainder is 2, so 3 is not a divisor of 1085)
...
1085 / 31 = 35 (the remainder is 0, so 31 and 35 are divisors of 1085)
1085 / 32 = 33.90625 (the remainder is 29, so 32 is not a divisor of 1085)