What are the divisors of 1356?

1, 2, 3, 4, 6, 12, 113, 226, 339, 452, 678, 1356

There is a total of 12 positive divisors.
The sum of these divisors is 3192.
The arithmetic mean is 266.

8 even divisors

2, 4, 6, 12, 226, 452, 678, 1356

4 odd divisors

1, 3, 113, 339

How to compute the divisors of 1356?

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 1356 by each of the numbers from 1 to 1356 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)

1356 / 1 = 1356 (the remainder is 0, so 1 is a divisor of 1356)
1356 / 2 = 678 (the remainder is 0, so 2 is a divisor of 1356)
1356 / 3 = 452 (the remainder is 0, so 3 is a divisor of 1356)
...
1356 / 1355 = 1.0007380073801 (the remainder is 1, so 1355 is not a divisor of 1356)
1356 / 1356 = 1 (the remainder is 0, so 1356 is a divisor of 1356)

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 1356 (i.e. 36.823905279044). 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 $\frac{N}{d} = D$ )

1356 / 1 = 1356 (the remainder is 0, so 1 and 1356 are divisors of 1356)
1356 / 2 = 678 (the remainder is 0, so 2 and 678 are divisors of 1356)
1356 / 3 = 452 (the remainder is 0, so 3 and 452 are divisors of 1356)
...
1356 / 35 = 38.742857142857 (the remainder is 26, so 35 is not a divisor of 1356)
1356 / 36 = 37.666666666667 (the remainder is 24, so 36 is not a divisor of 1356)