What are the divisors of 1029?

1, 3, 7, 21, 49, 147, 343, 1029

There is a total of 8 positive divisors.
The sum of these divisors is 1600.
The arithmetic mean is 200.

8 odd divisors

1, 3, 7, 21, 49, 147, 343, 1029

How to compute the divisors of 1029?

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 1029 by each of the numbers from 1 to 1029 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)

1029 / 1 = 1029 (the remainder is 0, so 1 is a divisor of 1029)
1029 / 2 = 514.5 (the remainder is 1, so 2 is not a divisor of 1029)
1029 / 3 = 343 (the remainder is 0, so 3 is a divisor of 1029)
...
1029 / 1028 = 1.0009727626459 (the remainder is 1, so 1028 is not a divisor of 1029)
1029 / 1029 = 1 (the remainder is 0, so 1029 is a divisor of 1029)

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 1029 (i.e. 32.078029864691). 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 $\frac{N}{d} = D$ )

1029 / 1 = 1029 (the remainder is 0, so 1 and 1029 are divisors of 1029)
1029 / 2 = 514.5 (the remainder is 1, so 2 is not a divisor of 1029)
1029 / 3 = 343 (the remainder is 0, so 3 and 343 are divisors of 1029)
...
1029 / 31 = 33.193548387097 (the remainder is 6, so 31 is not a divisor of 1029)
1029 / 32 = 32.15625 (the remainder is 5, so 32 is not a divisor of 1029)