What are the divisors of 105?

1, 3, 5, 7, 15, 21, 35, 105

There is a total of 8 positive divisors.
The sum of these divisors is 192.
The arithmetic mean is 24.

8 odd divisors

1, 3, 5, 7, 15, 21, 35, 105

How to compute the divisors of 105?

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 105 by each of the numbers from 1 to 105 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)

105 / 1 = 105 (the remainder is 0, so 1 is a divisor of 105)
105 / 2 = 52.5 (the remainder is 1, so 2 is not a divisor of 105)
105 / 3 = 35 (the remainder is 0, so 3 is a divisor of 105)
...
105 / 104 = 1.0096153846154 (the remainder is 1, so 104 is not a divisor of 105)
105 / 105 = 1 (the remainder is 0, so 105 is a divisor of 105)

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 105 (i.e. 10.24695076596). 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 $\frac{N}{d} = D$ )

105 / 1 = 105 (the remainder is 0, so 1 and 105 are divisors of 105)
105 / 2 = 52.5 (the remainder is 1, so 2 is not a divisor of 105)
105 / 3 = 35 (the remainder is 0, so 3 and 35 are divisors of 105)
...
105 / 9 = 11.666666666667 (the remainder is 6, so 9 is not a divisor of 105)
105 / 10 = 10.5 (the remainder is 5, so 10 is not a divisor of 105)