What are the divisors of 1233?

1, 3, 9, 137, 411, 1233

There is a total of 6 positive divisors.
The sum of these divisors is 1794.
The arithmetic mean is 299.

6 odd divisors

1, 3, 9, 137, 411, 1233

How to compute the divisors of 1233?

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 1233 by each of the numbers from 1 to 1233 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)

1233 / 1 = 1233 (the remainder is 0, so 1 is a divisor of 1233)
1233 / 2 = 616.5 (the remainder is 1, so 2 is not a divisor of 1233)
1233 / 3 = 411 (the remainder is 0, so 3 is a divisor of 1233)
...
1233 / 1232 = 1.0008116883117 (the remainder is 1, so 1232 is not a divisor of 1233)
1233 / 1233 = 1 (the remainder is 0, so 1233 is a divisor of 1233)

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 1233 (i.e. 35.114099732159). 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 $\frac{N}{d} = D$ )

1233 / 1 = 1233 (the remainder is 0, so 1 and 1233 are divisors of 1233)
1233 / 2 = 616.5 (the remainder is 1, so 2 is not a divisor of 1233)
1233 / 3 = 411 (the remainder is 0, so 3 and 411 are divisors of 1233)
...
1233 / 34 = 36.264705882353 (the remainder is 9, so 34 is not a divisor of 1233)
1233 / 35 = 35.228571428571 (the remainder is 8, so 35 is not a divisor of 1233)