What are the divisors of 1259?

1, 1259

There is a total of 2 positive divisors.
The sum of these divisors is 1260.
The arithmetic mean is 630.

2 odd divisors

1, 1259

How to compute the divisors of 1259?

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 1259 by each of the numbers from 1 to 1259 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)

1259 / 1 = 1259 (the remainder is 0, so 1 is a divisor of 1259)
1259 / 2 = 629.5 (the remainder is 1, so 2 is not a divisor of 1259)
1259 / 3 = 419.66666666667 (the remainder is 2, so 3 is not a divisor of 1259)
...
1259 / 1258 = 1.0007949125596 (the remainder is 1, so 1258 is not a divisor of 1259)
1259 / 1259 = 1 (the remainder is 0, so 1259 is a divisor of 1259)

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 1259 (i.e. 35.48238999842). 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 $\frac{N}{d} = D$ )

1259 / 1 = 1259 (the remainder is 0, so 1 and 1259 are divisors of 1259)
1259 / 2 = 629.5 (the remainder is 1, so 2 is not a divisor of 1259)
1259 / 3 = 419.66666666667 (the remainder is 2, so 3 is not a divisor of 1259)
...
1259 / 34 = 37.029411764706 (the remainder is 1, so 34 is not a divisor of 1259)
1259 / 35 = 35.971428571429 (the remainder is 34, so 35 is not a divisor of 1259)