What are the divisors of 1054?

1, 2, 17, 31, 34, 62, 527, 1054

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

4 even divisors

2, 34, 62, 1054

4 odd divisors

1, 17, 31, 527

How to compute the divisors of 1054?

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 1054 by each of the numbers from 1 to 1054 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)

1054 / 1 = 1054 (the remainder is 0, so 1 is a divisor of 1054)
1054 / 2 = 527 (the remainder is 0, so 2 is a divisor of 1054)
1054 / 3 = 351.33333333333 (the remainder is 1, so 3 is not a divisor of 1054)
...
1054 / 1053 = 1.0009496676163 (the remainder is 1, so 1053 is not a divisor of 1054)
1054 / 1054 = 1 (the remainder is 0, so 1054 is a divisor of 1054)

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 1054 (i.e. 32.465366161496). 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 $\frac{N}{d} = D$ )

1054 / 1 = 1054 (the remainder is 0, so 1 and 1054 are divisors of 1054)
1054 / 2 = 527 (the remainder is 0, so 2 and 527 are divisors of 1054)
1054 / 3 = 351.33333333333 (the remainder is 1, so 3 is not a divisor of 1054)
...
1054 / 31 = 34 (the remainder is 0, so 31 and 34 are divisors of 1054)
1054 / 32 = 32.9375 (the remainder is 30, so 32 is not a divisor of 1054)