What are the divisors of 1614?

1, 2, 3, 6, 269, 538, 807, 1614

There is a total of 8 positive divisors.
The sum of these divisors is 3240.
The arithmetic mean is 405.

4 even divisors

2, 6, 538, 1614

4 odd divisors

1, 3, 269, 807

How to compute the divisors of 1614?

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 1614 by each of the numbers from 1 to 1614 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)

1614 / 1 = 1614 (the remainder is 0, so 1 is a divisor of 1614)
1614 / 2 = 807 (the remainder is 0, so 2 is a divisor of 1614)
1614 / 3 = 538 (the remainder is 0, so 3 is a divisor of 1614)
...
1614 / 1613 = 1.0006199628022 (the remainder is 1, so 1613 is not a divisor of 1614)
1614 / 1614 = 1 (the remainder is 0, so 1614 is a divisor of 1614)

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 1614 (i.e. 40.174618853201). 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 $\frac{N}{d} = D$ )

1614 / 1 = 1614 (the remainder is 0, so 1 and 1614 are divisors of 1614)
1614 / 2 = 807 (the remainder is 0, so 2 and 807 are divisors of 1614)
1614 / 3 = 538 (the remainder is 0, so 3 and 538 are divisors of 1614)
...
1614 / 39 = 41.384615384615 (the remainder is 15, so 39 is not a divisor of 1614)
1614 / 40 = 40.35 (the remainder is 14, so 40 is not a divisor of 1614)