What are the divisors of 1378?

1, 2, 13, 26, 53, 106, 689, 1378

There is a total of 8 positive divisors.
The sum of these divisors is 2268.
The arithmetic mean is 283.5.

4 even divisors

2, 26, 106, 1378

4 odd divisors

1, 13, 53, 689

How to compute the divisors of 1378?

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 1378 by each of the numbers from 1 to 1378 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)

1378 / 1 = 1378 (the remainder is 0, so 1 is a divisor of 1378)
1378 / 2 = 689 (the remainder is 0, so 2 is a divisor of 1378)
1378 / 3 = 459.33333333333 (the remainder is 1, so 3 is not a divisor of 1378)
...
1378 / 1377 = 1.0007262164125 (the remainder is 1, so 1377 is not a divisor of 1378)
1378 / 1378 = 1 (the remainder is 0, so 1378 is a divisor of 1378)

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 1378 (i.e. 37.121422386541). 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 $\frac{N}{d} = D$ )

1378 / 1 = 1378 (the remainder is 0, so 1 and 1378 are divisors of 1378)
1378 / 2 = 689 (the remainder is 0, so 2 and 689 are divisors of 1378)
1378 / 3 = 459.33333333333 (the remainder is 1, so 3 is not a divisor of 1378)
...
1378 / 36 = 38.277777777778 (the remainder is 10, so 36 is not a divisor of 1378)
1378 / 37 = 37.243243243243 (the remainder is 9, so 37 is not a divisor of 1378)