What are the divisors of 101?

1, 101

There is a total of 2 positive divisors.
The sum of these divisors is 102.
The arithmetic mean is 51.

2 odd divisors

1, 101

How to compute the divisors of 101?

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 101 by each of the numbers from 1 to 101 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)

101 / 1 = 101 (the remainder is 0, so 1 is a divisor of 101)
101 / 2 = 50.5 (the remainder is 1, so 2 is not a divisor of 101)
101 / 3 = 33.666666666667 (the remainder is 2, so 3 is not a divisor of 101)
...
101 / 100 = 1.01 (the remainder is 1, so 100 is not a divisor of 101)
101 / 101 = 1 (the remainder is 0, so 101 is a divisor of 101)

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 101 (i.e. 10.049875621121). 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 $\frac{N}{d} = D$ )

101 / 1 = 101 (the remainder is 0, so 1 and 101 are divisors of 101)
101 / 2 = 50.5 (the remainder is 1, so 2 is not a divisor of 101)
101 / 3 = 33.666666666667 (the remainder is 2, so 3 is not a divisor of 101)
...
101 / 9 = 11.222222222222 (the remainder is 2, so 9 is not a divisor of 101)
101 / 10 = 10.1 (the remainder is 1, so 10 is not a divisor of 101)