What are the divisors of 431?

1, 431

There is a total of 2 positive divisors.
The sum of these divisors is 432.
The arithmetic mean is 216.

2 odd divisors

1, 431

How to compute the divisors of 431?

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 431 by each of the numbers from 1 to 431 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)

431 / 1 = 431 (the remainder is 0, so 1 is a divisor of 431)
431 / 2 = 215.5 (the remainder is 1, so 2 is not a divisor of 431)
431 / 3 = 143.66666666667 (the remainder is 2, so 3 is not a divisor of 431)
...
431 / 430 = 1.0023255813953 (the remainder is 1, so 430 is not a divisor of 431)
431 / 431 = 1 (the remainder is 0, so 431 is a divisor of 431)

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 431 (i.e. 20.760539492027). 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 $\frac{N}{d} = D$ )

431 / 1 = 431 (the remainder is 0, so 1 and 431 are divisors of 431)
431 / 2 = 215.5 (the remainder is 1, so 2 is not a divisor of 431)
431 / 3 = 143.66666666667 (the remainder is 2, so 3 is not a divisor of 431)
...
431 / 19 = 22.684210526316 (the remainder is 13, so 19 is not a divisor of 431)
431 / 20 = 21.55 (the remainder is 11, so 20 is not a divisor of 431)