What are the divisors of 1033?

1, 1033

There is a total of 2 positive divisors.
The sum of these divisors is 1034.
The arithmetic mean is 517.

2 odd divisors

1, 1033

How to compute the divisors of 1033?

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 1033 by each of the numbers from 1 to 1033 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)

1033 / 1 = 1033 (the remainder is 0, so 1 is a divisor of 1033)
1033 / 2 = 516.5 (the remainder is 1, so 2 is not a divisor of 1033)
1033 / 3 = 344.33333333333 (the remainder is 1, so 3 is not a divisor of 1033)
...
1033 / 1032 = 1.0009689922481 (the remainder is 1, so 1032 is not a divisor of 1033)
1033 / 1033 = 1 (the remainder is 0, so 1033 is a divisor of 1033)

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 1033 (i.e. 32.140317359976). 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 $\frac{N}{d} = D$ )

1033 / 1 = 1033 (the remainder is 0, so 1 and 1033 are divisors of 1033)
1033 / 2 = 516.5 (the remainder is 1, so 2 is not a divisor of 1033)
1033 / 3 = 344.33333333333 (the remainder is 1, so 3 is not a divisor of 1033)
...
1033 / 31 = 33.322580645161 (the remainder is 10, so 31 is not a divisor of 1033)
1033 / 32 = 32.28125 (the remainder is 9, so 32 is not a divisor of 1033)