What are the divisors of 1209?

1, 3, 13, 31, 39, 93, 403, 1209

There is a total of 8 positive divisors.
The sum of these divisors is 1792.
The arithmetic mean is 224.

8 odd divisors

1, 3, 13, 31, 39, 93, 403, 1209

How to compute the divisors of 1209?

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 1209 by each of the numbers from 1 to 1209 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)

1209 / 1 = 1209 (the remainder is 0, so 1 is a divisor of 1209)
1209 / 2 = 604.5 (the remainder is 1, so 2 is not a divisor of 1209)
1209 / 3 = 403 (the remainder is 0, so 3 is a divisor of 1209)
...
1209 / 1208 = 1.0008278145695 (the remainder is 1, so 1208 is not a divisor of 1209)
1209 / 1209 = 1 (the remainder is 0, so 1209 is a divisor of 1209)

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 1209 (i.e. 34.770677301427). 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 $\frac{N}{d} = D$ )

1209 / 1 = 1209 (the remainder is 0, so 1 and 1209 are divisors of 1209)
1209 / 2 = 604.5 (the remainder is 1, so 2 is not a divisor of 1209)
1209 / 3 = 403 (the remainder is 0, so 3 and 403 are divisors of 1209)
...
1209 / 33 = 36.636363636364 (the remainder is 21, so 33 is not a divisor of 1209)
1209 / 34 = 35.558823529412 (the remainder is 19, so 34 is not a divisor of 1209)