What are the divisors of 103?

1, 103

There is a total of 2 positive divisors.
The sum of these divisors is 104.
The arithmetic mean is 52.

2 odd divisors

1, 103

How to compute the divisors of 103?

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 103 by each of the numbers from 1 to 103 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)

103 / 1 = 103 (the remainder is 0, so 1 is a divisor of 103)
103 / 2 = 51.5 (the remainder is 1, so 2 is not a divisor of 103)
103 / 3 = 34.333333333333 (the remainder is 1, so 3 is not a divisor of 103)
...
103 / 102 = 1.0098039215686 (the remainder is 1, so 102 is not a divisor of 103)
103 / 103 = 1 (the remainder is 0, so 103 is a divisor of 103)

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 103 (i.e. 10.148891565092). 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 $\frac{N}{d} = D$ )

103 / 1 = 103 (the remainder is 0, so 1 and 103 are divisors of 103)
103 / 2 = 51.5 (the remainder is 1, so 2 is not a divisor of 103)
103 / 3 = 34.333333333333 (the remainder is 1, so 3 is not a divisor of 103)
...
103 / 9 = 11.444444444444 (the remainder is 4, so 9 is not a divisor of 103)
103 / 10 = 10.3 (the remainder is 3, so 10 is not a divisor of 103)