What are the divisors of 1231?

1, 1231

There is a total of 2 positive divisors.
The sum of these divisors is 1232.
The arithmetic mean is 616.

2 odd divisors

1, 1231

How to compute the divisors of 1231?

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 1231 by each of the numbers from 1 to 1231 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)

1231 / 1 = 1231 (the remainder is 0, so 1 is a divisor of 1231)
1231 / 2 = 615.5 (the remainder is 1, so 2 is not a divisor of 1231)
1231 / 3 = 410.33333333333 (the remainder is 1, so 3 is not a divisor of 1231)
...
1231 / 1230 = 1.0008130081301 (the remainder is 1, so 1230 is not a divisor of 1231)
1231 / 1231 = 1 (the remainder is 0, so 1231 is a divisor of 1231)

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 1231 (i.e. 35.085609585698). 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 $\frac{N}{d} = D$ )

1231 / 1 = 1231 (the remainder is 0, so 1 and 1231 are divisors of 1231)
1231 / 2 = 615.5 (the remainder is 1, so 2 is not a divisor of 1231)
1231 / 3 = 410.33333333333 (the remainder is 1, so 3 is not a divisor of 1231)
...
1231 / 34 = 36.205882352941 (the remainder is 7, so 34 is not a divisor of 1231)
1231 / 35 = 35.171428571429 (the remainder is 6, so 35 is not a divisor of 1231)