What are the divisors of 253?

1, 11, 23, 253

There is a total of 4 positive divisors.
The sum of these divisors is 288.
The arithmetic mean is 72.

4 odd divisors

1, 11, 23, 253

How to compute the divisors of 253?

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 253 by each of the numbers from 1 to 253 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)

253 / 1 = 253 (the remainder is 0, so 1 is a divisor of 253)
253 / 2 = 126.5 (the remainder is 1, so 2 is not a divisor of 253)
253 / 3 = 84.333333333333 (the remainder is 1, so 3 is not a divisor of 253)
...
253 / 252 = 1.0039682539683 (the remainder is 1, so 252 is not a divisor of 253)
253 / 253 = 1 (the remainder is 0, so 253 is a divisor of 253)

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 253 (i.e. 15.905973720587). 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 $\frac{N}{d} = D$ )

253 / 1 = 253 (the remainder is 0, so 1 and 253 are divisors of 253)
253 / 2 = 126.5 (the remainder is 1, so 2 is not a divisor of 253)
253 / 3 = 84.333333333333 (the remainder is 1, so 3 is not a divisor of 253)
...
253 / 14 = 18.071428571429 (the remainder is 1, so 14 is not a divisor of 253)
253 / 15 = 16.866666666667 (the remainder is 13, so 15 is not a divisor of 253)