What are the divisors of 1091?

1, 1091

There is a total of 2 positive divisors.
The sum of these divisors is 1092.
The arithmetic mean is 546.

2 odd divisors

1, 1091

How to compute the divisors of 1091?

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 1091 by each of the numbers from 1 to 1091 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)

1091 / 1 = 1091 (the remainder is 0, so 1 is a divisor of 1091)
1091 / 2 = 545.5 (the remainder is 1, so 2 is not a divisor of 1091)
1091 / 3 = 363.66666666667 (the remainder is 2, so 3 is not a divisor of 1091)
...
1091 / 1090 = 1.0009174311927 (the remainder is 1, so 1090 is not a divisor of 1091)
1091 / 1091 = 1 (the remainder is 0, so 1091 is a divisor of 1091)

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 1091 (i.e. 33.030289129827). 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 $\frac{N}{d} = D$ )

1091 / 1 = 1091 (the remainder is 0, so 1 and 1091 are divisors of 1091)
1091 / 2 = 545.5 (the remainder is 1, so 2 is not a divisor of 1091)
1091 / 3 = 363.66666666667 (the remainder is 2, so 3 is not a divisor of 1091)
...
1091 / 32 = 34.09375 (the remainder is 3, so 32 is not a divisor of 1091)
1091 / 33 = 33.060606060606 (the remainder is 2, so 33 is not a divisor of 1091)