What are the divisors of 274?

1, 2, 137, 274

There is a total of 4 positive divisors.
The sum of these divisors is 414.
The arithmetic mean is 103.5.

2 even divisors

2, 274

2 odd divisors

1, 137

How to compute the divisors of 274?

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 274 by each of the numbers from 1 to 274 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)

274 / 1 = 274 (the remainder is 0, so 1 is a divisor of 274)
274 / 2 = 137 (the remainder is 0, so 2 is a divisor of 274)
274 / 3 = 91.333333333333 (the remainder is 1, so 3 is not a divisor of 274)
...
274 / 273 = 1.003663003663 (the remainder is 1, so 273 is not a divisor of 274)
274 / 274 = 1 (the remainder is 0, so 274 is a divisor of 274)

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 274 (i.e. 16.552945357247). 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 $\frac{N}{d} = D$ )

274 / 1 = 274 (the remainder is 0, so 1 and 274 are divisors of 274)
274 / 2 = 137 (the remainder is 0, so 2 and 137 are divisors of 274)
274 / 3 = 91.333333333333 (the remainder is 1, so 3 is not a divisor of 274)
...
274 / 15 = 18.266666666667 (the remainder is 4, so 15 is not a divisor of 274)
274 / 16 = 17.125 (the remainder is 2, so 16 is not a divisor of 274)