What are the divisors of 4051?

1, 4051

There is a total of 2 positive divisors.
The sum of these divisors is 4052.
The arithmetic mean is 2026.

2 odd divisors

1, 4051

How to compute the divisors of 4051?

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 4051 by each of the numbers from 1 to 4051 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)

4051 / 1 = 4051 (the remainder is 0, so 1 is a divisor of 4051)
4051 / 2 = 2025.5 (the remainder is 1, so 2 is not a divisor of 4051)
4051 / 3 = 1350.3333333333 (the remainder is 1, so 3 is not a divisor of 4051)
...
4051 / 4050 = 1.0002469135802 (the remainder is 1, so 4050 is not a divisor of 4051)
4051 / 4051 = 1 (the remainder is 0, so 4051 is a divisor of 4051)

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 4051 (i.e. 63.647466563878). 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 $\frac{N}{d} = D$ )

4051 / 1 = 4051 (the remainder is 0, so 1 and 4051 are divisors of 4051)
4051 / 2 = 2025.5 (the remainder is 1, so 2 is not a divisor of 4051)
4051 / 3 = 1350.3333333333 (the remainder is 1, so 3 is not a divisor of 4051)
...
4051 / 62 = 65.338709677419 (the remainder is 21, so 62 is not a divisor of 4051)
4051 / 63 = 64.301587301587 (the remainder is 19, so 63 is not a divisor of 4051)