What are the divisors of 159?

1, 3, 53, 159

There is a total of 4 positive divisors.
The sum of these divisors is 216.
The arithmetic mean is 54.

4 odd divisors

1, 3, 53, 159

How to compute the divisors of 159?

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 159 by each of the numbers from 1 to 159 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)

159 / 1 = 159 (the remainder is 0, so 1 is a divisor of 159)
159 / 2 = 79.5 (the remainder is 1, so 2 is not a divisor of 159)
159 / 3 = 53 (the remainder is 0, so 3 is a divisor of 159)
...
159 / 158 = 1.0063291139241 (the remainder is 1, so 158 is not a divisor of 159)
159 / 159 = 1 (the remainder is 0, so 159 is a divisor of 159)

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 159 (i.e. 12.609520212918). 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 $\frac{N}{d} = D$ )

159 / 1 = 159 (the remainder is 0, so 1 and 159 are divisors of 159)
159 / 2 = 79.5 (the remainder is 1, so 2 is not a divisor of 159)
159 / 3 = 53 (the remainder is 0, so 3 and 53 are divisors of 159)
...
159 / 11 = 14.454545454545 (the remainder is 5, so 11 is not a divisor of 159)
159 / 12 = 13.25 (the remainder is 3, so 12 is not a divisor of 159)