What are the divisors of 3535?

1, 5, 7, 35, 101, 505, 707, 3535

There is a total of 8 positive divisors.
The sum of these divisors is 4896.
The arithmetic mean is 612.

8 odd divisors

1, 5, 7, 35, 101, 505, 707, 3535

How to compute the divisors of 3535?

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 3535 by each of the numbers from 1 to 3535 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)

3535 / 1 = 3535 (the remainder is 0, so 1 is a divisor of 3535)
3535 / 2 = 1767.5 (the remainder is 1, so 2 is not a divisor of 3535)
3535 / 3 = 1178.3333333333 (the remainder is 1, so 3 is not a divisor of 3535)
...
3535 / 3534 = 1.0002829654782 (the remainder is 1, so 3534 is not a divisor of 3535)
3535 / 3535 = 1 (the remainder is 0, so 3535 is a divisor of 3535)

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 3535 (i.e. 59.455865984779). 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 $\frac{N}{d} = D$ )

3535 / 1 = 3535 (the remainder is 0, so 1 and 3535 are divisors of 3535)
3535 / 2 = 1767.5 (the remainder is 1, so 2 is not a divisor of 3535)
3535 / 3 = 1178.3333333333 (the remainder is 1, so 3 is not a divisor of 3535)
...
3535 / 58 = 60.948275862069 (the remainder is 55, so 58 is not a divisor of 3535)
3535 / 59 = 59.915254237288 (the remainder is 54, so 59 is not a divisor of 3535)