What are the divisors of 1469?

1, 13, 113, 1469

There is a total of 4 positive divisors.
The sum of these divisors is 1596.
The arithmetic mean is 399.

4 odd divisors

1, 13, 113, 1469

How to compute the divisors of 1469?

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 1469 by each of the numbers from 1 to 1469 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)

1469 / 1 = 1469 (the remainder is 0, so 1 is a divisor of 1469)
1469 / 2 = 734.5 (the remainder is 1, so 2 is not a divisor of 1469)
1469 / 3 = 489.66666666667 (the remainder is 2, so 3 is not a divisor of 1469)
...
1469 / 1468 = 1.0006811989101 (the remainder is 1, so 1468 is not a divisor of 1469)
1469 / 1469 = 1 (the remainder is 0, so 1469 is a divisor of 1469)

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 1469 (i.e. 38.327535793474). 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 $\frac{N}{d} = D$ )

1469 / 1 = 1469 (the remainder is 0, so 1 and 1469 are divisors of 1469)
1469 / 2 = 734.5 (the remainder is 1, so 2 is not a divisor of 1469)
1469 / 3 = 489.66666666667 (the remainder is 2, so 3 is not a divisor of 1469)
...
1469 / 37 = 39.702702702703 (the remainder is 26, so 37 is not a divisor of 1469)
1469 / 38 = 38.657894736842 (the remainder is 25, so 38 is not a divisor of 1469)