What are the divisors of 1362?

1, 2, 3, 6, 227, 454, 681, 1362

There is a total of 8 positive divisors.
The sum of these divisors is 2736.
The arithmetic mean is 342.

4 even divisors

2, 6, 454, 1362

4 odd divisors

1, 3, 227, 681

How to compute the divisors of 1362?

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 1362 by each of the numbers from 1 to 1362 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)

1362 / 1 = 1362 (the remainder is 0, so 1 is a divisor of 1362)
1362 / 2 = 681 (the remainder is 0, so 2 is a divisor of 1362)
1362 / 3 = 454 (the remainder is 0, so 3 is a divisor of 1362)
...
1362 / 1361 = 1.0007347538575 (the remainder is 1, so 1361 is not a divisor of 1362)
1362 / 1362 = 1 (the remainder is 0, so 1362 is a divisor of 1362)

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 1362 (i.e. 36.905284174492). 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 $\frac{N}{d} = D$ )

1362 / 1 = 1362 (the remainder is 0, so 1 and 1362 are divisors of 1362)
1362 / 2 = 681 (the remainder is 0, so 2 and 681 are divisors of 1362)
1362 / 3 = 454 (the remainder is 0, so 3 and 454 are divisors of 1362)
...
1362 / 35 = 38.914285714286 (the remainder is 32, so 35 is not a divisor of 1362)
1362 / 36 = 37.833333333333 (the remainder is 30, so 36 is not a divisor of 1362)