What are the divisors of 1067?

1, 11, 97, 1067

There is a total of 4 positive divisors.
The sum of these divisors is 1176.
The arithmetic mean is 294.

4 odd divisors

1, 11, 97, 1067

How to compute the divisors of 1067?

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 1067 by each of the numbers from 1 to 1067 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)

1067 / 1 = 1067 (the remainder is 0, so 1 is a divisor of 1067)
1067 / 2 = 533.5 (the remainder is 1, so 2 is not a divisor of 1067)
1067 / 3 = 355.66666666667 (the remainder is 2, so 3 is not a divisor of 1067)
...
1067 / 1066 = 1.0009380863039 (the remainder is 1, so 1066 is not a divisor of 1067)
1067 / 1067 = 1 (the remainder is 0, so 1067 is a divisor of 1067)

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 1067 (i.e. 32.664965942122). 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 $\frac{N}{d} = D$ )

1067 / 1 = 1067 (the remainder is 0, so 1 and 1067 are divisors of 1067)
1067 / 2 = 533.5 (the remainder is 1, so 2 is not a divisor of 1067)
1067 / 3 = 355.66666666667 (the remainder is 2, so 3 is not a divisor of 1067)
...
1067 / 31 = 34.41935483871 (the remainder is 13, so 31 is not a divisor of 1067)
1067 / 32 = 33.34375 (the remainder is 11, so 32 is not a divisor of 1067)