What are the divisors of 1113?

1, 3, 7, 21, 53, 159, 371, 1113

There is a total of 8 positive divisors.
The sum of these divisors is 1728.
The arithmetic mean is 216.

8 odd divisors

1, 3, 7, 21, 53, 159, 371, 1113

How to compute the divisors of 1113?

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 1113 by each of the numbers from 1 to 1113 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)

1113 / 1 = 1113 (the remainder is 0, so 1 is a divisor of 1113)
1113 / 2 = 556.5 (the remainder is 1, so 2 is not a divisor of 1113)
1113 / 3 = 371 (the remainder is 0, so 3 is a divisor of 1113)
...
1113 / 1112 = 1.0008992805755 (the remainder is 1, so 1112 is not a divisor of 1113)
1113 / 1113 = 1 (the remainder is 0, so 1113 is a divisor of 1113)

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 1113 (i.e. 33.361654635225). 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 $\frac{N}{d} = D$ )

1113 / 1 = 1113 (the remainder is 0, so 1 and 1113 are divisors of 1113)
1113 / 2 = 556.5 (the remainder is 1, so 2 is not a divisor of 1113)
1113 / 3 = 371 (the remainder is 0, so 3 and 371 are divisors of 1113)
...
1113 / 32 = 34.78125 (the remainder is 25, so 32 is not a divisor of 1113)
1113 / 33 = 33.727272727273 (the remainder is 24, so 33 is not a divisor of 1113)