What are the divisors of 413?

1, 7, 59, 413

There is a total of 4 positive divisors.
The sum of these divisors is 480.
The arithmetic mean is 120.

4 odd divisors

1, 7, 59, 413

How to compute the divisors of 413?

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 413 by each of the numbers from 1 to 413 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)

413 / 1 = 413 (the remainder is 0, so 1 is a divisor of 413)
413 / 2 = 206.5 (the remainder is 1, so 2 is not a divisor of 413)
413 / 3 = 137.66666666667 (the remainder is 2, so 3 is not a divisor of 413)
...
413 / 412 = 1.002427184466 (the remainder is 1, so 412 is not a divisor of 413)
413 / 413 = 1 (the remainder is 0, so 413 is a divisor of 413)

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 413 (i.e. 20.322401432902). 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 $\frac{N}{d} = D$ )

413 / 1 = 413 (the remainder is 0, so 1 and 413 are divisors of 413)
413 / 2 = 206.5 (the remainder is 1, so 2 is not a divisor of 413)
413 / 3 = 137.66666666667 (the remainder is 2, so 3 is not a divisor of 413)
...
413 / 19 = 21.736842105263 (the remainder is 14, so 19 is not a divisor of 413)
413 / 20 = 20.65 (the remainder is 13, so 20 is not a divisor of 413)