What are the divisors of 1613?

1, 1613

There is a total of 2 positive divisors.
The sum of these divisors is 1614.
The arithmetic mean is 807.

2 odd divisors

1, 1613

How to compute the divisors of 1613?

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 1613 by each of the numbers from 1 to 1613 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)

1613 / 1 = 1613 (the remainder is 0, so 1 is a divisor of 1613)
1613 / 2 = 806.5 (the remainder is 1, so 2 is not a divisor of 1613)
1613 / 3 = 537.66666666667 (the remainder is 2, so 3 is not a divisor of 1613)
...
1613 / 1612 = 1.0006203473945 (the remainder is 1, so 1612 is not a divisor of 1613)
1613 / 1613 = 1 (the remainder is 0, so 1613 is a divisor of 1613)

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 1613 (i.e. 40.162171256046). 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 $\frac{N}{d} = D$ )

1613 / 1 = 1613 (the remainder is 0, so 1 and 1613 are divisors of 1613)
1613 / 2 = 806.5 (the remainder is 1, so 2 is not a divisor of 1613)
1613 / 3 = 537.66666666667 (the remainder is 2, so 3 is not a divisor of 1613)
...
1613 / 39 = 41.358974358974 (the remainder is 14, so 39 is not a divisor of 1613)
1613 / 40 = 40.325 (the remainder is 13, so 40 is not a divisor of 1613)