What are the divisors of 3503?

1, 31, 113, 3503

There is a total of 4 positive divisors.
The sum of these divisors is 3648.
The arithmetic mean is 912.

4 odd divisors

1, 31, 113, 3503

How to compute the divisors of 3503?

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 3503 by each of the numbers from 1 to 3503 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)

3503 / 1 = 3503 (the remainder is 0, so 1 is a divisor of 3503)
3503 / 2 = 1751.5 (the remainder is 1, so 2 is not a divisor of 3503)
3503 / 3 = 1167.6666666667 (the remainder is 2, so 3 is not a divisor of 3503)
...
3503 / 3502 = 1.0002855511136 (the remainder is 1, so 3502 is not a divisor of 3503)
3503 / 3503 = 1 (the remainder is 0, so 3503 is a divisor of 3503)

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 3503 (i.e. 59.186147027831). 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 $\frac{N}{d} = D$ )

3503 / 1 = 3503 (the remainder is 0, so 1 and 3503 are divisors of 3503)
3503 / 2 = 1751.5 (the remainder is 1, so 2 is not a divisor of 3503)
3503 / 3 = 1167.6666666667 (the remainder is 2, so 3 is not a divisor of 3503)
...
3503 / 58 = 60.396551724138 (the remainder is 23, so 58 is not a divisor of 3503)
3503 / 59 = 59.372881355932 (the remainder is 22, so 59 is not a divisor of 3503)