What are the divisors of 445?

1, 5, 89, 445

There is a total of 4 positive divisors.
The sum of these divisors is 540.
The arithmetic mean is 135.

4 odd divisors

1, 5, 89, 445

How to compute the divisors of 445?

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 445 by each of the numbers from 1 to 445 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)

445 / 1 = 445 (the remainder is 0, so 1 is a divisor of 445)
445 / 2 = 222.5 (the remainder is 1, so 2 is not a divisor of 445)
445 / 3 = 148.33333333333 (the remainder is 1, so 3 is not a divisor of 445)
...
445 / 444 = 1.0022522522523 (the remainder is 1, so 444 is not a divisor of 445)
445 / 445 = 1 (the remainder is 0, so 445 is a divisor of 445)

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 445 (i.e. 21.095023109729). 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 $\frac{N}{d} = D$ )

445 / 1 = 445 (the remainder is 0, so 1 and 445 are divisors of 445)
445 / 2 = 222.5 (the remainder is 1, so 2 is not a divisor of 445)
445 / 3 = 148.33333333333 (the remainder is 1, so 3 is not a divisor of 445)
...
445 / 20 = 22.25 (the remainder is 5, so 20 is not a divisor of 445)
445 / 21 = 21.190476190476 (the remainder is 4, so 21 is not a divisor of 445)