What are the divisors of 1074?

1, 2, 3, 6, 179, 358, 537, 1074

There is a total of 8 positive divisors.
The sum of these divisors is 2160.
The arithmetic mean is 270.

4 even divisors

2, 6, 358, 1074

4 odd divisors

1, 3, 179, 537

How to compute the divisors of 1074?

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 1074 by each of the numbers from 1 to 1074 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)

1074 / 1 = 1074 (the remainder is 0, so 1 is a divisor of 1074)
1074 / 2 = 537 (the remainder is 0, so 2 is a divisor of 1074)
1074 / 3 = 358 (the remainder is 0, so 3 is a divisor of 1074)
...
1074 / 1073 = 1.0009319664492 (the remainder is 1, so 1073 is not a divisor of 1074)
1074 / 1074 = 1 (the remainder is 0, so 1074 is a divisor of 1074)

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 1074 (i.e. 32.771939216348). 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 $\frac{N}{d} = D$ )

1074 / 1 = 1074 (the remainder is 0, so 1 and 1074 are divisors of 1074)
1074 / 2 = 537 (the remainder is 0, so 2 and 537 are divisors of 1074)
1074 / 3 = 358 (the remainder is 0, so 3 and 358 are divisors of 1074)
...
1074 / 31 = 34.645161290323 (the remainder is 20, so 31 is not a divisor of 1074)
1074 / 32 = 33.5625 (the remainder is 18, so 32 is not a divisor of 1074)