What are the divisors of 435?

1, 3, 5, 15, 29, 87, 145, 435

There is a total of 8 positive divisors.
The sum of these divisors is 720.
The arithmetic mean is 90.

8 odd divisors

1, 3, 5, 15, 29, 87, 145, 435

How to compute the divisors of 435?

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 435 by each of the numbers from 1 to 435 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)

435 / 1 = 435 (the remainder is 0, so 1 is a divisor of 435)
435 / 2 = 217.5 (the remainder is 1, so 2 is not a divisor of 435)
435 / 3 = 145 (the remainder is 0, so 3 is a divisor of 435)
...
435 / 434 = 1.0023041474654 (the remainder is 1, so 434 is not a divisor of 435)
435 / 435 = 1 (the remainder is 0, so 435 is a divisor of 435)

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 435 (i.e. 20.856653614614). 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 $\frac{N}{d} = D$ )

435 / 1 = 435 (the remainder is 0, so 1 and 435 are divisors of 435)
435 / 2 = 217.5 (the remainder is 1, so 2 is not a divisor of 435)
435 / 3 = 145 (the remainder is 0, so 3 and 145 are divisors of 435)
...
435 / 19 = 22.894736842105 (the remainder is 17, so 19 is not a divisor of 435)
435 / 20 = 21.75 (the remainder is 15, so 20 is not a divisor of 435)