What are the divisors of 241?

1, 241

There is a total of 2 positive divisors.
The sum of these divisors is 242.
The arithmetic mean is 121.

2 odd divisors

1, 241

How to compute the divisors of 241?

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 241 by each of the numbers from 1 to 241 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)

241 / 1 = 241 (the remainder is 0, so 1 is a divisor of 241)
241 / 2 = 120.5 (the remainder is 1, so 2 is not a divisor of 241)
241 / 3 = 80.333333333333 (the remainder is 1, so 3 is not a divisor of 241)
...
241 / 240 = 1.0041666666667 (the remainder is 1, so 240 is not a divisor of 241)
241 / 241 = 1 (the remainder is 0, so 241 is a divisor of 241)

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 241 (i.e. 15.52417469626). 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 $\frac{N}{d} = D$ )

241 / 1 = 241 (the remainder is 0, so 1 and 241 are divisors of 241)
241 / 2 = 120.5 (the remainder is 1, so 2 is not a divisor of 241)
241 / 3 = 80.333333333333 (the remainder is 1, so 3 is not a divisor of 241)
...
241 / 14 = 17.214285714286 (the remainder is 3, so 14 is not a divisor of 241)
241 / 15 = 16.066666666667 (the remainder is 1, so 15 is not a divisor of 241)