What are the divisors of 119?

1, 7, 17, 119

There is a total of 4 positive divisors.
The sum of these divisors is 144.
The arithmetic mean is 36.

4 odd divisors

1, 7, 17, 119

How to compute the divisors of 119?

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 119 by each of the numbers from 1 to 119 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)

119 / 1 = 119 (the remainder is 0, so 1 is a divisor of 119)
119 / 2 = 59.5 (the remainder is 1, so 2 is not a divisor of 119)
119 / 3 = 39.666666666667 (the remainder is 2, so 3 is not a divisor of 119)
...
119 / 118 = 1.0084745762712 (the remainder is 1, so 118 is not a divisor of 119)
119 / 119 = 1 (the remainder is 0, so 119 is a divisor of 119)

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 119 (i.e. 10.908712114636). 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 $\frac{N}{d} = D$ )

119 / 1 = 119 (the remainder is 0, so 1 and 119 are divisors of 119)
119 / 2 = 59.5 (the remainder is 1, so 2 is not a divisor of 119)
119 / 3 = 39.666666666667 (the remainder is 2, so 3 is not a divisor of 119)
...
119 / 9 = 13.222222222222 (the remainder is 2, so 9 is not a divisor of 119)
119 / 10 = 11.9 (the remainder is 9, so 10 is not a divisor of 119)