What are the divisors of 2027?

1, 2027

There is a total of 2 positive divisors.
The sum of these divisors is 2028.
The arithmetic mean is 1014.

2 odd divisors

1, 2027

How to compute the divisors of 2027?

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 2027 by each of the numbers from 1 to 2027 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)

2027 / 1 = 2027 (the remainder is 0, so 1 is a divisor of 2027)
2027 / 2 = 1013.5 (the remainder is 1, so 2 is not a divisor of 2027)
2027 / 3 = 675.66666666667 (the remainder is 2, so 3 is not a divisor of 2027)
...
2027 / 2026 = 1.0004935834156 (the remainder is 1, so 2026 is not a divisor of 2027)
2027 / 2027 = 1 (the remainder is 0, so 2027 is a divisor of 2027)

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 2027 (i.e. 45.022216737962). 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 $\frac{N}{d} = D$ )

2027 / 1 = 2027 (the remainder is 0, so 1 and 2027 are divisors of 2027)
2027 / 2 = 1013.5 (the remainder is 1, so 2 is not a divisor of 2027)
2027 / 3 = 675.66666666667 (the remainder is 2, so 3 is not a divisor of 2027)
...
2027 / 44 = 46.068181818182 (the remainder is 3, so 44 is not a divisor of 2027)
2027 / 45 = 45.044444444444 (the remainder is 2, so 45 is not a divisor of 2027)