What are the divisors of 427?

1, 7, 61, 427

There is a total of 4 positive divisors.
The sum of these divisors is 496.
The arithmetic mean is 124.

4 odd divisors

1, 7, 61, 427

How to compute the divisors of 427?

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 427 by each of the numbers from 1 to 427 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)

427 / 1 = 427 (the remainder is 0, so 1 is a divisor of 427)
427 / 2 = 213.5 (the remainder is 1, so 2 is not a divisor of 427)
427 / 3 = 142.33333333333 (the remainder is 1, so 3 is not a divisor of 427)
...
427 / 426 = 1.0023474178404 (the remainder is 1, so 426 is not a divisor of 427)
427 / 427 = 1 (the remainder is 0, so 427 is a divisor of 427)

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 427 (i.e. 20.663978319772). 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 $\frac{N}{d} = D$ )

427 / 1 = 427 (the remainder is 0, so 1 and 427 are divisors of 427)
427 / 2 = 213.5 (the remainder is 1, so 2 is not a divisor of 427)
427 / 3 = 142.33333333333 (the remainder is 1, so 3 is not a divisor of 427)
...
427 / 19 = 22.473684210526 (the remainder is 9, so 19 is not a divisor of 427)
427 / 20 = 21.35 (the remainder is 7, so 20 is not a divisor of 427)