What are the divisors of 174?

1, 2, 3, 6, 29, 58, 87, 174

There is a total of 8 positive divisors.
The sum of these divisors is 360.
The arithmetic mean is 45.

4 even divisors

2, 6, 58, 174

4 odd divisors

1, 3, 29, 87

How to compute the divisors of 174?

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 174 by each of the numbers from 1 to 174 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)

174 / 1 = 174 (the remainder is 0, so 1 is a divisor of 174)
174 / 2 = 87 (the remainder is 0, so 2 is a divisor of 174)
174 / 3 = 58 (the remainder is 0, so 3 is a divisor of 174)
...
174 / 173 = 1.0057803468208 (the remainder is 1, so 173 is not a divisor of 174)
174 / 174 = 1 (the remainder is 0, so 174 is a divisor of 174)

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 174 (i.e. 13.190905958273). 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 $\frac{N}{d} = D$ )

174 / 1 = 174 (the remainder is 0, so 1 and 174 are divisors of 174)
174 / 2 = 87 (the remainder is 0, so 2 and 87 are divisors of 174)
174 / 3 = 58 (the remainder is 0, so 3 and 58 are divisors of 174)
...
174 / 12 = 14.5 (the remainder is 6, so 12 is not a divisor of 174)
174 / 13 = 13.384615384615 (the remainder is 5, so 13 is not a divisor of 174)