What are the divisors of 403?

1, 13, 31, 403

There is a total of 4 positive divisors.
The sum of these divisors is 448.
The arithmetic mean is 112.

4 odd divisors

1, 13, 31, 403

How to compute the divisors of 403?

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 403 by each of the numbers from 1 to 403 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)

403 / 1 = 403 (the remainder is 0, so 1 is a divisor of 403)
403 / 2 = 201.5 (the remainder is 1, so 2 is not a divisor of 403)
403 / 3 = 134.33333333333 (the remainder is 1, so 3 is not a divisor of 403)
...
403 / 402 = 1.0024875621891 (the remainder is 1, so 402 is not a divisor of 403)
403 / 403 = 1 (the remainder is 0, so 403 is a divisor of 403)

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 403 (i.e. 20.074859899885). 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 $\frac{N}{d} = D$ )

403 / 1 = 403 (the remainder is 0, so 1 and 403 are divisors of 403)
403 / 2 = 201.5 (the remainder is 1, so 2 is not a divisor of 403)
403 / 3 = 134.33333333333 (the remainder is 1, so 3 is not a divisor of 403)
...
403 / 19 = 21.210526315789 (the remainder is 4, so 19 is not a divisor of 403)
403 / 20 = 20.15 (the remainder is 3, so 20 is not a divisor of 403)