What are the divisors of 1266?

1, 2, 3, 6, 211, 422, 633, 1266

There is a total of 8 positive divisors.
The sum of these divisors is 2544.
The arithmetic mean is 318.

4 even divisors

2, 6, 422, 1266

4 odd divisors

1, 3, 211, 633

How to compute the divisors of 1266?

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 1266 by each of the numbers from 1 to 1266 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)

1266 / 1 = 1266 (the remainder is 0, so 1 is a divisor of 1266)
1266 / 2 = 633 (the remainder is 0, so 2 is a divisor of 1266)
1266 / 3 = 422 (the remainder is 0, so 3 is a divisor of 1266)
...
1266 / 1265 = 1.000790513834 (the remainder is 1, so 1265 is not a divisor of 1266)
1266 / 1266 = 1 (the remainder is 0, so 1266 is a divisor of 1266)

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 1266 (i.e. 35.580893749314). 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 $\frac{N}{d} = D$ )

1266 / 1 = 1266 (the remainder is 0, so 1 and 1266 are divisors of 1266)
1266 / 2 = 633 (the remainder is 0, so 2 and 633 are divisors of 1266)
1266 / 3 = 422 (the remainder is 0, so 3 and 422 are divisors of 1266)
...
1266 / 34 = 37.235294117647 (the remainder is 8, so 34 is not a divisor of 1266)
1266 / 35 = 36.171428571429 (the remainder is 6, so 35 is not a divisor of 1266)