What are the divisors of 1053?

1, 3, 9, 13, 27, 39, 81, 117, 351, 1053

There is a total of 10 positive divisors.
The sum of these divisors is 1694.
The arithmetic mean is 169.4.

10 odd divisors

1, 3, 9, 13, 27, 39, 81, 117, 351, 1053

How to compute the divisors of 1053?

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 1053 by each of the numbers from 1 to 1053 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)

1053 / 1 = 1053 (the remainder is 0, so 1 is a divisor of 1053)
1053 / 2 = 526.5 (the remainder is 1, so 2 is not a divisor of 1053)
1053 / 3 = 351 (the remainder is 0, so 3 is a divisor of 1053)
...
1053 / 1052 = 1.0009505703422 (the remainder is 1, so 1052 is not a divisor of 1053)
1053 / 1053 = 1 (the remainder is 0, so 1053 is a divisor of 1053)

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 1053 (i.e. 32.449961479176). 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 $\frac{N}{d} = D$ )

1053 / 1 = 1053 (the remainder is 0, so 1 and 1053 are divisors of 1053)
1053 / 2 = 526.5 (the remainder is 1, so 2 is not a divisor of 1053)
1053 / 3 = 351 (the remainder is 0, so 3 and 351 are divisors of 1053)
...
1053 / 31 = 33.967741935484 (the remainder is 30, so 31 is not a divisor of 1053)
1053 / 32 = 32.90625 (the remainder is 29, so 32 is not a divisor of 1053)