What are the divisors of 1254?

1, 2, 3, 6, 11, 19, 22, 33, 38, 57, 66, 114, 209, 418, 627, 1254

There is a total of 16 positive divisors.
The sum of these divisors is 2880.
The arithmetic mean is 180.

8 even divisors

2, 6, 22, 38, 66, 114, 418, 1254

8 odd divisors

1, 3, 11, 19, 33, 57, 209, 627

How to compute the divisors of 1254?

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 1254 by each of the numbers from 1 to 1254 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)

1254 / 1 = 1254 (the remainder is 0, so 1 is a divisor of 1254)
1254 / 2 = 627 (the remainder is 0, so 2 is a divisor of 1254)
1254 / 3 = 418 (the remainder is 0, so 3 is a divisor of 1254)
...
1254 / 1253 = 1.000798084597 (the remainder is 1, so 1253 is not a divisor of 1254)
1254 / 1254 = 1 (the remainder is 0, so 1254 is a divisor of 1254)

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 1254 (i.e. 35.411862419252). 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 $\frac{N}{d} = D$ )

1254 / 1 = 1254 (the remainder is 0, so 1 and 1254 are divisors of 1254)
1254 / 2 = 627 (the remainder is 0, so 2 and 627 are divisors of 1254)
1254 / 3 = 418 (the remainder is 0, so 3 and 418 are divisors of 1254)
...
1254 / 34 = 36.882352941176 (the remainder is 30, so 34 is not a divisor of 1254)
1254 / 35 = 35.828571428571 (the remainder is 29, so 35 is not a divisor of 1254)