What are the divisors of 1413?

1, 3, 9, 157, 471, 1413

There is a total of 6 positive divisors.
The sum of these divisors is 2054.
The arithmetic mean is 342.33333333333.

6 odd divisors

1, 3, 9, 157, 471, 1413

How to compute the divisors of 1413?

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 1413 by each of the numbers from 1 to 1413 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)

1413 / 1 = 1413 (the remainder is 0, so 1 is a divisor of 1413)
1413 / 2 = 706.5 (the remainder is 1, so 2 is not a divisor of 1413)
1413 / 3 = 471 (the remainder is 0, so 3 is a divisor of 1413)
...
1413 / 1412 = 1.0007082152975 (the remainder is 1, so 1412 is not a divisor of 1413)
1413 / 1413 = 1 (the remainder is 0, so 1413 is a divisor of 1413)

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 1413 (i.e. 37.589892258425). 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 $\frac{N}{d} = D$ )

1413 / 1 = 1413 (the remainder is 0, so 1 and 1413 are divisors of 1413)
1413 / 2 = 706.5 (the remainder is 1, so 2 is not a divisor of 1413)
1413 / 3 = 471 (the remainder is 0, so 3 and 471 are divisors of 1413)
...
1413 / 36 = 39.25 (the remainder is 9, so 36 is not a divisor of 1413)
1413 / 37 = 38.189189189189 (the remainder is 7, so 37 is not a divisor of 1413)