What are the divisors of 4007?

1, 4007

There is a total of 2 positive divisors.
The sum of these divisors is 4008.
The arithmetic mean is 2004.

2 odd divisors

1, 4007

How to compute the divisors of 4007?

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 4007 by each of the numbers from 1 to 4007 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)

4007 / 1 = 4007 (the remainder is 0, so 1 is a divisor of 4007)
4007 / 2 = 2003.5 (the remainder is 1, so 2 is not a divisor of 4007)
4007 / 3 = 1335.6666666667 (the remainder is 2, so 3 is not a divisor of 4007)
...
4007 / 4006 = 1.0002496255617 (the remainder is 1, so 4006 is not a divisor of 4007)
4007 / 4007 = 1 (the remainder is 0, so 4007 is a divisor of 4007)

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 4007 (i.e. 63.300868872394). 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 $\frac{N}{d} = D$ )

4007 / 1 = 4007 (the remainder is 0, so 1 and 4007 are divisors of 4007)
4007 / 2 = 2003.5 (the remainder is 1, so 2 is not a divisor of 4007)
4007 / 3 = 1335.6666666667 (the remainder is 2, so 3 is not a divisor of 4007)
...
4007 / 62 = 64.629032258065 (the remainder is 39, so 62 is not a divisor of 4007)
4007 / 63 = 63.603174603175 (the remainder is 38, so 63 is not a divisor of 4007)