What are the divisors of 1109?

1, 1109

There is a total of 2 positive divisors.
The sum of these divisors is 1110.
The arithmetic mean is 555.

2 odd divisors

1, 1109

How to compute the divisors of 1109?

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 1109 by each of the numbers from 1 to 1109 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)

1109 / 1 = 1109 (the remainder is 0, so 1 is a divisor of 1109)
1109 / 2 = 554.5 (the remainder is 1, so 2 is not a divisor of 1109)
1109 / 3 = 369.66666666667 (the remainder is 2, so 3 is not a divisor of 1109)
...
1109 / 1108 = 1.0009025270758 (the remainder is 1, so 1108 is not a divisor of 1109)
1109 / 1109 = 1 (the remainder is 0, so 1109 is a divisor of 1109)

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 1109 (i.e. 33.301651610693). 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 $\frac{N}{d} = D$ )

1109 / 1 = 1109 (the remainder is 0, so 1 and 1109 are divisors of 1109)
1109 / 2 = 554.5 (the remainder is 1, so 2 is not a divisor of 1109)
1109 / 3 = 369.66666666667 (the remainder is 2, so 3 is not a divisor of 1109)
...
1109 / 32 = 34.65625 (the remainder is 21, so 32 is not a divisor of 1109)
1109 / 33 = 33.606060606061 (the remainder is 20, so 33 is not a divisor of 1109)