What are the divisors of 128?

1, 2, 4, 8, 16, 32, 64, 128

There is a total of 8 positive divisors.
The sum of these divisors is 255.
The arithmetic mean is 31.875.

7 even divisors

2, 4, 8, 16, 32, 64, 128

1 odd divisors

How to compute the divisors of 128?

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 128 by each of the numbers from 1 to 128 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)

128 / 1 = 128 (the remainder is 0, so 1 is a divisor of 128)
128 / 2 = 64 (the remainder is 0, so 2 is a divisor of 128)
128 / 3 = 42.666666666667 (the remainder is 2, so 3 is not a divisor of 128)
...
128 / 127 = 1.007874015748 (the remainder is 1, so 127 is not a divisor of 128)
128 / 128 = 1 (the remainder is 0, so 128 is a divisor of 128)

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 128 (i.e. 11.313708498985). 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 $\frac{N}{d} = D$ )

128 / 1 = 128 (the remainder is 0, so 1 and 128 are divisors of 128)
128 / 2 = 64 (the remainder is 0, so 2 and 64 are divisors of 128)
128 / 3 = 42.666666666667 (the remainder is 2, so 3 is not a divisor of 128)
...
128 / 10 = 12.8 (the remainder is 8, so 10 is not a divisor of 128)
128 / 11 = 11.636363636364 (the remainder is 7, so 11 is not a divisor of 128)