What are the divisors of 1267?

1, 7, 181, 1267

There is a total of 4 positive divisors.
The sum of these divisors is 1456.
The arithmetic mean is 364.

4 odd divisors

1, 7, 181, 1267

How to compute the divisors of 1267?

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 1267 by each of the numbers from 1 to 1267 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)

1267 / 1 = 1267 (the remainder is 0, so 1 is a divisor of 1267)
1267 / 2 = 633.5 (the remainder is 1, so 2 is not a divisor of 1267)
1267 / 3 = 422.33333333333 (the remainder is 1, so 3 is not a divisor of 1267)
...
1267 / 1266 = 1.0007898894155 (the remainder is 1, so 1266 is not a divisor of 1267)
1267 / 1267 = 1 (the remainder is 0, so 1267 is a divisor of 1267)

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 1267 (i.e. 35.594943461115). 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 $\frac{N}{d} = D$ )

1267 / 1 = 1267 (the remainder is 0, so 1 and 1267 are divisors of 1267)
1267 / 2 = 633.5 (the remainder is 1, so 2 is not a divisor of 1267)
1267 / 3 = 422.33333333333 (the remainder is 1, so 3 is not a divisor of 1267)
...
1267 / 34 = 37.264705882353 (the remainder is 9, so 34 is not a divisor of 1267)
1267 / 35 = 36.2 (the remainder is 7, so 35 is not a divisor of 1267)