What are the divisors of 1251?

1, 3, 9, 139, 417, 1251

There is a total of 6 positive divisors.
The sum of these divisors is 1820.
The arithmetic mean is 303.33333333333.

6 odd divisors

1, 3, 9, 139, 417, 1251

How to compute the divisors of 1251?

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 1251 by each of the numbers from 1 to 1251 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)

1251 / 1 = 1251 (the remainder is 0, so 1 is a divisor of 1251)
1251 / 2 = 625.5 (the remainder is 1, so 2 is not a divisor of 1251)
1251 / 3 = 417 (the remainder is 0, so 3 is a divisor of 1251)
...
1251 / 1250 = 1.0008 (the remainder is 1, so 1250 is not a divisor of 1251)
1251 / 1251 = 1 (the remainder is 0, so 1251 is a divisor of 1251)

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 1251 (i.e. 35.369478367655). 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 $\frac{N}{d} = D$ )

1251 / 1 = 1251 (the remainder is 0, so 1 and 1251 are divisors of 1251)
1251 / 2 = 625.5 (the remainder is 1, so 2 is not a divisor of 1251)
1251 / 3 = 417 (the remainder is 0, so 3 and 417 are divisors of 1251)
...
1251 / 34 = 36.794117647059 (the remainder is 27, so 34 is not a divisor of 1251)
1251 / 35 = 35.742857142857 (the remainder is 26, so 35 is not a divisor of 1251)