What are the divisors of 1374?

1, 2, 3, 6, 229, 458, 687, 1374

There is a total of 8 positive divisors.
The sum of these divisors is 2760.
The arithmetic mean is 345.

4 even divisors

2, 6, 458, 1374

4 odd divisors

1, 3, 229, 687

How to compute the divisors of 1374?

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 1374 by each of the numbers from 1 to 1374 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)

1374 / 1 = 1374 (the remainder is 0, so 1 is a divisor of 1374)
1374 / 2 = 687 (the remainder is 0, so 2 is a divisor of 1374)
1374 / 3 = 458 (the remainder is 0, so 3 is a divisor of 1374)
...
1374 / 1373 = 1.0007283321194 (the remainder is 1, so 1373 is not a divisor of 1374)
1374 / 1374 = 1 (the remainder is 0, so 1374 is a divisor of 1374)

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 1374 (i.e. 37.067505985701). 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 $\frac{N}{d} = D$ )

1374 / 1 = 1374 (the remainder is 0, so 1 and 1374 are divisors of 1374)
1374 / 2 = 687 (the remainder is 0, so 2 and 687 are divisors of 1374)
1374 / 3 = 458 (the remainder is 0, so 3 and 458 are divisors of 1374)
...
1374 / 36 = 38.166666666667 (the remainder is 6, so 36 is not a divisor of 1374)
1374 / 37 = 37.135135135135 (the remainder is 5, so 37 is not a divisor of 1374)