What are the divisors of 449?

1, 449

There is a total of 2 positive divisors.
The sum of these divisors is 450.
The arithmetic mean is 225.

2 odd divisors

1, 449

How to compute the divisors of 449?

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 449 by each of the numbers from 1 to 449 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)

449 / 1 = 449 (the remainder is 0, so 1 is a divisor of 449)
449 / 2 = 224.5 (the remainder is 1, so 2 is not a divisor of 449)
449 / 3 = 149.66666666667 (the remainder is 2, so 3 is not a divisor of 449)
...
449 / 448 = 1.0022321428571 (the remainder is 1, so 448 is not a divisor of 449)
449 / 449 = 1 (the remainder is 0, so 449 is a divisor of 449)

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 449 (i.e. 21.189620100417). 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 $\frac{N}{d} = D$ )

449 / 1 = 449 (the remainder is 0, so 1 and 449 are divisors of 449)
449 / 2 = 224.5 (the remainder is 1, so 2 is not a divisor of 449)
449 / 3 = 149.66666666667 (the remainder is 2, so 3 is not a divisor of 449)
...
449 / 20 = 22.45 (the remainder is 9, so 20 is not a divisor of 449)
449 / 21 = 21.380952380952 (the remainder is 8, so 21 is not a divisor of 449)