What are the divisors of 133?

1, 7, 19, 133

There is a total of 4 positive divisors.
The sum of these divisors is 160.
The arithmetic mean is 40.

4 odd divisors

1, 7, 19, 133

How to compute the divisors of 133?

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 133 by each of the numbers from 1 to 133 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)

133 / 1 = 133 (the remainder is 0, so 1 is a divisor of 133)
133 / 2 = 66.5 (the remainder is 1, so 2 is not a divisor of 133)
133 / 3 = 44.333333333333 (the remainder is 1, so 3 is not a divisor of 133)
...
133 / 132 = 1.0075757575758 (the remainder is 1, so 132 is not a divisor of 133)
133 / 133 = 1 (the remainder is 0, so 133 is a divisor of 133)

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 133 (i.e. 11.532562594671). 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 $\frac{N}{d} = D$ )

133 / 1 = 133 (the remainder is 0, so 1 and 133 are divisors of 133)
133 / 2 = 66.5 (the remainder is 1, so 2 is not a divisor of 133)
133 / 3 = 44.333333333333 (the remainder is 1, so 3 is not a divisor of 133)
...
133 / 10 = 13.3 (the remainder is 3, so 10 is not a divisor of 133)
133 / 11 = 12.090909090909 (the remainder is 1, so 11 is not a divisor of 133)