What are the divisors of 457?

1, 457

There is a total of 2 positive divisors.
The sum of these divisors is 458.
The arithmetic mean is 229.

2 odd divisors

1, 457

How to compute the divisors of 457?

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 457 by each of the numbers from 1 to 457 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)

457 / 1 = 457 (the remainder is 0, so 1 is a divisor of 457)
457 / 2 = 228.5 (the remainder is 1, so 2 is not a divisor of 457)
457 / 3 = 152.33333333333 (the remainder is 1, so 3 is not a divisor of 457)
...
457 / 456 = 1.0021929824561 (the remainder is 1, so 456 is not a divisor of 457)
457 / 457 = 1 (the remainder is 0, so 457 is a divisor of 457)

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 457 (i.e. 21.377558326432). 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 $\frac{N}{d} = D$ )

457 / 1 = 457 (the remainder is 0, so 1 and 457 are divisors of 457)
457 / 2 = 228.5 (the remainder is 1, so 2 is not a divisor of 457)
457 / 3 = 152.33333333333 (the remainder is 1, so 3 is not a divisor of 457)
...
457 / 20 = 22.85 (the remainder is 17, so 20 is not a divisor of 457)
457 / 21 = 21.761904761905 (the remainder is 16, so 21 is not a divisor of 457)