What are the divisors of 1269?

1, 3, 9, 27, 47, 141, 423, 1269

There is a total of 8 positive divisors.
The sum of these divisors is 1920.
The arithmetic mean is 240.

8 odd divisors

1, 3, 9, 27, 47, 141, 423, 1269

How to compute the divisors of 1269?

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 1269 by each of the numbers from 1 to 1269 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)

1269 / 1 = 1269 (the remainder is 0, so 1 is a divisor of 1269)
1269 / 2 = 634.5 (the remainder is 1, so 2 is not a divisor of 1269)
1269 / 3 = 423 (the remainder is 0, so 3 is a divisor of 1269)
...
1269 / 1268 = 1.0007886435331 (the remainder is 1, so 1268 is not a divisor of 1269)
1269 / 1269 = 1 (the remainder is 0, so 1269 is a divisor of 1269)

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 1269 (i.e. 35.623026261114). 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 $\frac{N}{d} = D$ )

1269 / 1 = 1269 (the remainder is 0, so 1 and 1269 are divisors of 1269)
1269 / 2 = 634.5 (the remainder is 1, so 2 is not a divisor of 1269)
1269 / 3 = 423 (the remainder is 0, so 3 and 423 are divisors of 1269)
...
1269 / 34 = 37.323529411765 (the remainder is 11, so 34 is not a divisor of 1269)
1269 / 35 = 36.257142857143 (the remainder is 9, so 35 is not a divisor of 1269)