What are the divisors of 338?

1, 2, 13, 26, 169, 338

There is a total of 6 positive divisors.
The sum of these divisors is 549.
The arithmetic mean is 91.5.

3 even divisors

2, 26, 338

3 odd divisors

1, 13, 169

How to compute the divisors of 338?

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 338 by each of the numbers from 1 to 338 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)

338 / 1 = 338 (the remainder is 0, so 1 is a divisor of 338)
338 / 2 = 169 (the remainder is 0, so 2 is a divisor of 338)
338 / 3 = 112.66666666667 (the remainder is 2, so 3 is not a divisor of 338)
...
338 / 337 = 1.0029673590504 (the remainder is 1, so 337 is not a divisor of 338)
338 / 338 = 1 (the remainder is 0, so 338 is a divisor of 338)

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 338 (i.e. 18.38477631085). 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 $\frac{N}{d} = D$ )

338 / 1 = 338 (the remainder is 0, so 1 and 338 are divisors of 338)
338 / 2 = 169 (the remainder is 0, so 2 and 169 are divisors of 338)
338 / 3 = 112.66666666667 (the remainder is 2, so 3 is not a divisor of 338)
...
338 / 17 = 19.882352941176 (the remainder is 15, so 17 is not a divisor of 338)
338 / 18 = 18.777777777778 (the remainder is 14, so 18 is not a divisor of 338)