What are the divisors of 263?

1, 263

There is a total of 2 positive divisors.
The sum of these divisors is 264.
The arithmetic mean is 132.

2 odd divisors

1, 263

How to compute the divisors of 263?

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 263 by each of the numbers from 1 to 263 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)

263 / 1 = 263 (the remainder is 0, so 1 is a divisor of 263)
263 / 2 = 131.5 (the remainder is 1, so 2 is not a divisor of 263)
263 / 3 = 87.666666666667 (the remainder is 2, so 3 is not a divisor of 263)
...
263 / 262 = 1.0038167938931 (the remainder is 1, so 262 is not a divisor of 263)
263 / 263 = 1 (the remainder is 0, so 263 is a divisor of 263)

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 263 (i.e. 16.217274740227). 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 $\frac{N}{d} = D$ )

263 / 1 = 263 (the remainder is 0, so 1 and 263 are divisors of 263)
263 / 2 = 131.5 (the remainder is 1, so 2 is not a divisor of 263)
263 / 3 = 87.666666666667 (the remainder is 2, so 3 is not a divisor of 263)
...
263 / 15 = 17.533333333333 (the remainder is 8, so 15 is not a divisor of 263)
263 / 16 = 16.4375 (the remainder is 7, so 16 is not a divisor of 263)