What are the divisors of 2233?

1, 7, 11, 29, 77, 203, 319, 2233

There is a total of 8 positive divisors.
The sum of these divisors is 2880.
The arithmetic mean is 360.

8 odd divisors

1, 7, 11, 29, 77, 203, 319, 2233

How to compute the divisors of 2233?

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 2233 by each of the numbers from 1 to 2233 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)

2233 / 1 = 2233 (the remainder is 0, so 1 is a divisor of 2233)
2233 / 2 = 1116.5 (the remainder is 1, so 2 is not a divisor of 2233)
2233 / 3 = 744.33333333333 (the remainder is 1, so 3 is not a divisor of 2233)
...
2233 / 2232 = 1.0004480286738 (the remainder is 1, so 2232 is not a divisor of 2233)
2233 / 2233 = 1 (the remainder is 0, so 2233 is a divisor of 2233)

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 2233 (i.e. 47.254629402843). 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 $\frac{N}{d} = D$ )

2233 / 1 = 2233 (the remainder is 0, so 1 and 2233 are divisors of 2233)
2233 / 2 = 1116.5 (the remainder is 1, so 2 is not a divisor of 2233)
2233 / 3 = 744.33333333333 (the remainder is 1, so 3 is not a divisor of 2233)
...
2233 / 46 = 48.54347826087 (the remainder is 25, so 46 is not a divisor of 2233)
2233 / 47 = 47.510638297872 (the remainder is 24, so 47 is not a divisor of 2233)