What are the divisors of 1013?

1, 1013

There is a total of 2 positive divisors.
The sum of these divisors is 1014.
The arithmetic mean is 507.

2 odd divisors

1, 1013

How to compute the divisors of 1013?

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 1013 by each of the numbers from 1 to 1013 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)

1013 / 1 = 1013 (the remainder is 0, so 1 is a divisor of 1013)
1013 / 2 = 506.5 (the remainder is 1, so 2 is not a divisor of 1013)
1013 / 3 = 337.66666666667 (the remainder is 2, so 3 is not a divisor of 1013)
...
1013 / 1012 = 1.0009881422925 (the remainder is 1, so 1012 is not a divisor of 1013)
1013 / 1013 = 1 (the remainder is 0, so 1013 is a divisor of 1013)

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 1013 (i.e. 31.827660925679). 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 $\frac{N}{d} = D$ )

1013 / 1 = 1013 (the remainder is 0, so 1 and 1013 are divisors of 1013)
1013 / 2 = 506.5 (the remainder is 1, so 2 is not a divisor of 1013)
1013 / 3 = 337.66666666667 (the remainder is 2, so 3 is not a divisor of 1013)
...
1013 / 30 = 33.766666666667 (the remainder is 23, so 30 is not a divisor of 1013)
1013 / 31 = 32.677419354839 (the remainder is 21, so 31 is not a divisor of 1013)