What are the divisors of 1249?

1, 1249

There is a total of 2 positive divisors.
The sum of these divisors is 1250.
The arithmetic mean is 625.

2 odd divisors

1, 1249

How to compute the divisors of 1249?

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 1249 by each of the numbers from 1 to 1249 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)

1249 / 1 = 1249 (the remainder is 0, so 1 is a divisor of 1249)
1249 / 2 = 624.5 (the remainder is 1, so 2 is not a divisor of 1249)
1249 / 3 = 416.33333333333 (the remainder is 1, so 3 is not a divisor of 1249)
...
1249 / 1248 = 1.0008012820513 (the remainder is 1, so 1248 is not a divisor of 1249)
1249 / 1249 = 1 (the remainder is 0, so 1249 is a divisor of 1249)

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 1249 (i.e. 35.341194094145). 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 $\frac{N}{d} = D$ )

1249 / 1 = 1249 (the remainder is 0, so 1 and 1249 are divisors of 1249)
1249 / 2 = 624.5 (the remainder is 1, so 2 is not a divisor of 1249)
1249 / 3 = 416.33333333333 (the remainder is 1, so 3 is not a divisor of 1249)
...
1249 / 34 = 36.735294117647 (the remainder is 25, so 34 is not a divisor of 1249)
1249 / 35 = 35.685714285714 (the remainder is 24, so 35 is not a divisor of 1249)