What are the divisors of 127?

1, 127

There is a total of 2 positive divisors.
The sum of these divisors is 128.
The arithmetic mean is 64.

2 odd divisors

1, 127

How to compute the divisors of 127?

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 127 by each of the numbers from 1 to 127 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)

127 / 1 = 127 (the remainder is 0, so 1 is a divisor of 127)
127 / 2 = 63.5 (the remainder is 1, so 2 is not a divisor of 127)
127 / 3 = 42.333333333333 (the remainder is 1, so 3 is not a divisor of 127)
...
127 / 126 = 1.0079365079365 (the remainder is 1, so 126 is not a divisor of 127)
127 / 127 = 1 (the remainder is 0, so 127 is a divisor of 127)

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 127 (i.e. 11.269427669585). 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 $\frac{N}{d} = D$ )

127 / 1 = 127 (the remainder is 0, so 1 and 127 are divisors of 127)
127 / 2 = 63.5 (the remainder is 1, so 2 is not a divisor of 127)
127 / 3 = 42.333333333333 (the remainder is 1, so 3 is not a divisor of 127)
...
127 / 10 = 12.7 (the remainder is 7, so 10 is not a divisor of 127)
127 / 11 = 11.545454545455 (the remainder is 6, so 11 is not a divisor of 127)