What are the divisors of 421?

1, 421

There is a total of 2 positive divisors.
The sum of these divisors is 422.
The arithmetic mean is 211.

2 odd divisors

1, 421

How to compute the divisors of 421?

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 421 by each of the numbers from 1 to 421 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)

421 / 1 = 421 (the remainder is 0, so 1 is a divisor of 421)
421 / 2 = 210.5 (the remainder is 1, so 2 is not a divisor of 421)
421 / 3 = 140.33333333333 (the remainder is 1, so 3 is not a divisor of 421)
...
421 / 420 = 1.002380952381 (the remainder is 1, so 420 is not a divisor of 421)
421 / 421 = 1 (the remainder is 0, so 421 is a divisor of 421)

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 421 (i.e. 20.518284528683). 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 $\frac{N}{d} = D$ )

421 / 1 = 421 (the remainder is 0, so 1 and 421 are divisors of 421)
421 / 2 = 210.5 (the remainder is 1, so 2 is not a divisor of 421)
421 / 3 = 140.33333333333 (the remainder is 1, so 3 is not a divisor of 421)
...
421 / 19 = 22.157894736842 (the remainder is 3, so 19 is not a divisor of 421)
421 / 20 = 21.05 (the remainder is 1, so 20 is not a divisor of 421)