What are the divisors of 1010?

1, 2, 5, 10, 101, 202, 505, 1010

There is a total of 8 positive divisors.
The sum of these divisors is 1836.
The arithmetic mean is 229.5.

4 even divisors

2, 10, 202, 1010

4 odd divisors

1, 5, 101, 505

How to compute the divisors of 1010?

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 1010 by each of the numbers from 1 to 1010 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)

1010 / 1 = 1010 (the remainder is 0, so 1 is a divisor of 1010)
1010 / 2 = 505 (the remainder is 0, so 2 is a divisor of 1010)
1010 / 3 = 336.66666666667 (the remainder is 2, so 3 is not a divisor of 1010)
...
1010 / 1009 = 1.0009910802775 (the remainder is 1, so 1009 is not a divisor of 1010)
1010 / 1010 = 1 (the remainder is 0, so 1010 is a divisor of 1010)

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 1010 (i.e. 31.780497164141). 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 $\frac{N}{d} = D$ )

1010 / 1 = 1010 (the remainder is 0, so 1 and 1010 are divisors of 1010)
1010 / 2 = 505 (the remainder is 0, so 2 and 505 are divisors of 1010)
1010 / 3 = 336.66666666667 (the remainder is 2, so 3 is not a divisor of 1010)
...
1010 / 30 = 33.666666666667 (the remainder is 20, so 30 is not a divisor of 1010)
1010 / 31 = 32.58064516129 (the remainder is 18, so 31 is not a divisor of 1010)