What are the divisors of 242?

1, 2, 11, 22, 121, 242

There is a total of 6 positive divisors.
The sum of these divisors is 399.
The arithmetic mean is 66.5.

3 even divisors

2, 22, 242

3 odd divisors

1, 11, 121

How to compute the divisors of 242?

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 242 by each of the numbers from 1 to 242 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)

242 / 1 = 242 (the remainder is 0, so 1 is a divisor of 242)
242 / 2 = 121 (the remainder is 0, so 2 is a divisor of 242)
242 / 3 = 80.666666666667 (the remainder is 2, so 3 is not a divisor of 242)
...
242 / 241 = 1.0041493775934 (the remainder is 1, so 241 is not a divisor of 242)
242 / 242 = 1 (the remainder is 0, so 242 is a divisor of 242)

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 242 (i.e. 15.556349186104). 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 $\frac{N}{d} = D$ )

242 / 1 = 242 (the remainder is 0, so 1 and 242 are divisors of 242)
242 / 2 = 121 (the remainder is 0, so 2 and 121 are divisors of 242)
242 / 3 = 80.666666666667 (the remainder is 2, so 3 is not a divisor of 242)
...
242 / 14 = 17.285714285714 (the remainder is 4, so 14 is not a divisor of 242)
242 / 15 = 16.133333333333 (the remainder is 2, so 15 is not a divisor of 242)