What are the divisors of 113?

1, 113

There is a total of 2 positive divisors.
The sum of these divisors is 114.
The arithmetic mean is 57.

2 odd divisors

1, 113

How to compute the divisors of 113?

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 113 by each of the numbers from 1 to 113 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)

113 / 1 = 113 (the remainder is 0, so 1 is a divisor of 113)
113 / 2 = 56.5 (the remainder is 1, so 2 is not a divisor of 113)
113 / 3 = 37.666666666667 (the remainder is 2, so 3 is not a divisor of 113)
...
113 / 112 = 1.0089285714286 (the remainder is 1, so 112 is not a divisor of 113)
113 / 113 = 1 (the remainder is 0, so 113 is a divisor of 113)

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 113 (i.e. 10.630145812735). 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 $\frac{N}{d} = D$ )

113 / 1 = 113 (the remainder is 0, so 1 and 113 are divisors of 113)
113 / 2 = 56.5 (the remainder is 1, so 2 is not a divisor of 113)
113 / 3 = 37.666666666667 (the remainder is 2, so 3 is not a divisor of 113)
...
113 / 9 = 12.555555555556 (the remainder is 5, so 9 is not a divisor of 113)
113 / 10 = 11.3 (the remainder is 3, so 10 is not a divisor of 113)