What are the divisors of 1121?

1, 19, 59, 1121

There is a total of 4 positive divisors.
The sum of these divisors is 1200.
The arithmetic mean is 300.

4 odd divisors

1, 19, 59, 1121

How to compute the divisors of 1121?

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 1121 by each of the numbers from 1 to 1121 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)

1121 / 1 = 1121 (the remainder is 0, so 1 is a divisor of 1121)
1121 / 2 = 560.5 (the remainder is 1, so 2 is not a divisor of 1121)
1121 / 3 = 373.66666666667 (the remainder is 2, so 3 is not a divisor of 1121)
...
1121 / 1120 = 1.0008928571429 (the remainder is 1, so 1120 is not a divisor of 1121)
1121 / 1121 = 1 (the remainder is 0, so 1121 is a divisor of 1121)

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 1121 (i.e. 33.481338085566). 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 $\frac{N}{d} = D$ )

1121 / 1 = 1121 (the remainder is 0, so 1 and 1121 are divisors of 1121)
1121 / 2 = 560.5 (the remainder is 1, so 2 is not a divisor of 1121)
1121 / 3 = 373.66666666667 (the remainder is 2, so 3 is not a divisor of 1121)
...
1121 / 32 = 35.03125 (the remainder is 1, so 32 is not a divisor of 1121)
1121 / 33 = 33.969696969697 (the remainder is 32, so 33 is not a divisor of 1121)