What are the divisors of 106?

1, 2, 53, 106

There is a total of 4 positive divisors.
The sum of these divisors is 162.
The arithmetic mean is 40.5.

2 even divisors

2, 106

2 odd divisors

1, 53

How to compute the divisors of 106?

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 106 by each of the numbers from 1 to 106 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)

106 / 1 = 106 (the remainder is 0, so 1 is a divisor of 106)
106 / 2 = 53 (the remainder is 0, so 2 is a divisor of 106)
106 / 3 = 35.333333333333 (the remainder is 1, so 3 is not a divisor of 106)
...
106 / 105 = 1.0095238095238 (the remainder is 1, so 105 is not a divisor of 106)
106 / 106 = 1 (the remainder is 0, so 106 is a divisor of 106)

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 106 (i.e. 10.295630140987). 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 $\frac{N}{d} = D$ )

106 / 1 = 106 (the remainder is 0, so 1 and 106 are divisors of 106)
106 / 2 = 53 (the remainder is 0, so 2 and 53 are divisors of 106)
106 / 3 = 35.333333333333 (the remainder is 1, so 3 is not a divisor of 106)
...
106 / 9 = 11.777777777778 (the remainder is 7, so 9 is not a divisor of 106)
106 / 10 = 10.6 (the remainder is 6, so 10 is not a divisor of 106)