What are the divisors of 249?

1, 3, 83, 249

There is a total of 4 positive divisors.
The sum of these divisors is 336.
The arithmetic mean is 84.

4 odd divisors

1, 3, 83, 249

How to compute the divisors of 249?

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 249 by each of the numbers from 1 to 249 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)

249 / 1 = 249 (the remainder is 0, so 1 is a divisor of 249)
249 / 2 = 124.5 (the remainder is 1, so 2 is not a divisor of 249)
249 / 3 = 83 (the remainder is 0, so 3 is a divisor of 249)
...
249 / 248 = 1.0040322580645 (the remainder is 1, so 248 is not a divisor of 249)
249 / 249 = 1 (the remainder is 0, so 249 is a divisor of 249)

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 249 (i.e. 15.779733838059). 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 $\frac{N}{d} = D$ )

249 / 1 = 249 (the remainder is 0, so 1 and 249 are divisors of 249)
249 / 2 = 124.5 (the remainder is 1, so 2 is not a divisor of 249)
249 / 3 = 83 (the remainder is 0, so 3 and 83 are divisors of 249)
...
249 / 14 = 17.785714285714 (the remainder is 11, so 14 is not a divisor of 249)
249 / 15 = 16.6 (the remainder is 9, so 15 is not a divisor of 249)