What are the divisors of 535?

1, 5, 107, 535

There is a total of 4 positive divisors.
The sum of these divisors is 648.
The arithmetic mean is 162.

4 odd divisors

1, 5, 107, 535

How to compute the divisors of 535?

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 535 by each of the numbers from 1 to 535 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)

535 / 1 = 535 (the remainder is 0, so 1 is a divisor of 535)
535 / 2 = 267.5 (the remainder is 1, so 2 is not a divisor of 535)
535 / 3 = 178.33333333333 (the remainder is 1, so 3 is not a divisor of 535)
...
535 / 534 = 1.001872659176 (the remainder is 1, so 534 is not a divisor of 535)
535 / 535 = 1 (the remainder is 0, so 535 is a divisor of 535)

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 535 (i.e. 23.130067012441). 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 $\frac{N}{d} = D$ )

535 / 1 = 535 (the remainder is 0, so 1 and 535 are divisors of 535)
535 / 2 = 267.5 (the remainder is 1, so 2 is not a divisor of 535)
535 / 3 = 178.33333333333 (the remainder is 1, so 3 is not a divisor of 535)
...
535 / 22 = 24.318181818182 (the remainder is 7, so 22 is not a divisor of 535)
535 / 23 = 23.260869565217 (the remainder is 6, so 23 is not a divisor of 535)