What are the divisors of 471?

1, 3, 157, 471

There is a total of 4 positive divisors.
The sum of these divisors is 632.
The arithmetic mean is 158.

4 odd divisors

1, 3, 157, 471

How to compute the divisors of 471?

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 471 by each of the numbers from 1 to 471 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)

471 / 1 = 471 (the remainder is 0, so 1 is a divisor of 471)
471 / 2 = 235.5 (the remainder is 1, so 2 is not a divisor of 471)
471 / 3 = 157 (the remainder is 0, so 3 is a divisor of 471)
...
471 / 470 = 1.0021276595745 (the remainder is 1, so 470 is not a divisor of 471)
471 / 471 = 1 (the remainder is 0, so 471 is a divisor of 471)

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 471 (i.e. 21.702534414211). 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 $\frac{N}{d} = D$ )

471 / 1 = 471 (the remainder is 0, so 1 and 471 are divisors of 471)
471 / 2 = 235.5 (the remainder is 1, so 2 is not a divisor of 471)
471 / 3 = 157 (the remainder is 0, so 3 and 157 are divisors of 471)
...
471 / 20 = 23.55 (the remainder is 11, so 20 is not a divisor of 471)
471 / 21 = 22.428571428571 (the remainder is 9, so 21 is not a divisor of 471)