What are the divisors of 1314?

1, 2, 3, 6, 9, 18, 73, 146, 219, 438, 657, 1314

There is a total of 12 positive divisors.
The sum of these divisors is 2886.
The arithmetic mean is 240.5.

6 even divisors

2, 6, 18, 146, 438, 1314

6 odd divisors

1, 3, 9, 73, 219, 657

How to compute the divisors of 1314?

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 1314 by each of the numbers from 1 to 1314 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)

1314 / 1 = 1314 (the remainder is 0, so 1 is a divisor of 1314)
1314 / 2 = 657 (the remainder is 0, so 2 is a divisor of 1314)
1314 / 3 = 438 (the remainder is 0, so 3 is a divisor of 1314)
...
1314 / 1313 = 1.000761614623 (the remainder is 1, so 1313 is not a divisor of 1314)
1314 / 1314 = 1 (the remainder is 0, so 1314 is a divisor of 1314)

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 1314 (i.e. 36.249137920784). 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 $\frac{N}{d} = D$ )

1314 / 1 = 1314 (the remainder is 0, so 1 and 1314 are divisors of 1314)
1314 / 2 = 657 (the remainder is 0, so 2 and 657 are divisors of 1314)
1314 / 3 = 438 (the remainder is 0, so 3 and 438 are divisors of 1314)
...
1314 / 35 = 37.542857142857 (the remainder is 19, so 35 is not a divisor of 1314)
1314 / 36 = 36.5 (the remainder is 18, so 36 is not a divisor of 1314)