What are the divisors of 1007?

1, 19, 53, 1007

There is a total of 4 positive divisors.
The sum of these divisors is 1080.
The arithmetic mean is 270.

4 odd divisors

1, 19, 53, 1007

How to compute the divisors of 1007?

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 1007 by each of the numbers from 1 to 1007 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)

1007 / 1 = 1007 (the remainder is 0, so 1 is a divisor of 1007)
1007 / 2 = 503.5 (the remainder is 1, so 2 is not a divisor of 1007)
1007 / 3 = 335.66666666667 (the remainder is 2, so 3 is not a divisor of 1007)
...
1007 / 1006 = 1.0009940357853 (the remainder is 1, so 1006 is not a divisor of 1007)
1007 / 1007 = 1 (the remainder is 0, so 1007 is a divisor of 1007)

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 1007 (i.e. 31.733263305245). 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 $\frac{N}{d} = D$ )

1007 / 1 = 1007 (the remainder is 0, so 1 and 1007 are divisors of 1007)
1007 / 2 = 503.5 (the remainder is 1, so 2 is not a divisor of 1007)
1007 / 3 = 335.66666666667 (the remainder is 2, so 3 is not a divisor of 1007)
...
1007 / 30 = 33.566666666667 (the remainder is 17, so 30 is not a divisor of 1007)
1007 / 31 = 32.483870967742 (the remainder is 15, so 31 is not a divisor of 1007)