What are the divisors of 1034?

1, 2, 11, 22, 47, 94, 517, 1034

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

4 even divisors

2, 22, 94, 1034

4 odd divisors

1, 11, 47, 517

How to compute the divisors of 1034?

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 1034 by each of the numbers from 1 to 1034 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)

1034 / 1 = 1034 (the remainder is 0, so 1 is a divisor of 1034)
1034 / 2 = 517 (the remainder is 0, so 2 is a divisor of 1034)
1034 / 3 = 344.66666666667 (the remainder is 2, so 3 is not a divisor of 1034)
...
1034 / 1033 = 1.000968054211 (the remainder is 1, so 1033 is not a divisor of 1034)
1034 / 1034 = 1 (the remainder is 0, so 1034 is a divisor of 1034)

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 1034 (i.e. 32.155870381627). 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 $\frac{N}{d} = D$ )

1034 / 1 = 1034 (the remainder is 0, so 1 and 1034 are divisors of 1034)
1034 / 2 = 517 (the remainder is 0, so 2 and 517 are divisors of 1034)
1034 / 3 = 344.66666666667 (the remainder is 2, so 3 is not a divisor of 1034)
...
1034 / 31 = 33.354838709677 (the remainder is 11, so 31 is not a divisor of 1034)
1034 / 32 = 32.3125 (the remainder is 10, so 32 is not a divisor of 1034)