What are the divisors of 1239?

1, 3, 7, 21, 59, 177, 413, 1239

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, 7, 21, 59, 177, 413, 1239

How to compute the divisors of 1239?

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 1239 by each of the numbers from 1 to 1239 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)

1239 / 1 = 1239 (the remainder is 0, so 1 is a divisor of 1239)
1239 / 2 = 619.5 (the remainder is 1, so 2 is not a divisor of 1239)
1239 / 3 = 413 (the remainder is 0, so 3 is a divisor of 1239)
...
1239 / 1238 = 1.0008077544426 (the remainder is 1, so 1238 is not a divisor of 1239)
1239 / 1239 = 1 (the remainder is 0, so 1239 is a divisor of 1239)

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 1239 (i.e. 35.199431813596). 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 $\frac{N}{d} = D$ )

1239 / 1 = 1239 (the remainder is 0, so 1 and 1239 are divisors of 1239)
1239 / 2 = 619.5 (the remainder is 1, so 2 is not a divisor of 1239)
1239 / 3 = 413 (the remainder is 0, so 3 and 413 are divisors of 1239)
...
1239 / 34 = 36.441176470588 (the remainder is 15, so 34 is not a divisor of 1239)
1239 / 35 = 35.4 (the remainder is 14, so 35 is not a divisor of 1239)