What are the divisors of 461?

1, 461

There is a total of 2 positive divisors.
The sum of these divisors is 462.
The arithmetic mean is 231.

2 odd divisors

1, 461

How to compute the divisors of 461?

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 461 by each of the numbers from 1 to 461 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)

461 / 1 = 461 (the remainder is 0, so 1 is a divisor of 461)
461 / 2 = 230.5 (the remainder is 1, so 2 is not a divisor of 461)
461 / 3 = 153.66666666667 (the remainder is 2, so 3 is not a divisor of 461)
...
461 / 460 = 1.0021739130435 (the remainder is 1, so 460 is not a divisor of 461)
461 / 461 = 1 (the remainder is 0, so 461 is a divisor of 461)

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 461 (i.e. 21.470910553584). 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 $\frac{N}{d} = D$ )

461 / 1 = 461 (the remainder is 0, so 1 and 461 are divisors of 461)
461 / 2 = 230.5 (the remainder is 1, so 2 is not a divisor of 461)
461 / 3 = 153.66666666667 (the remainder is 2, so 3 is not a divisor of 461)
...
461 / 20 = 23.05 (the remainder is 1, so 20 is not a divisor of 461)
461 / 21 = 21.952380952381 (the remainder is 20, so 21 is not a divisor of 461)