What are the divisors of 497?

1, 7, 71, 497

There is a total of 4 positive divisors.
The sum of these divisors is 576.
The arithmetic mean is 144.

4 odd divisors

1, 7, 71, 497

How to compute the divisors of 497?

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 497 by each of the numbers from 1 to 497 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)

497 / 1 = 497 (the remainder is 0, so 1 is a divisor of 497)
497 / 2 = 248.5 (the remainder is 1, so 2 is not a divisor of 497)
497 / 3 = 165.66666666667 (the remainder is 2, so 3 is not a divisor of 497)
...
497 / 496 = 1.0020161290323 (the remainder is 1, so 496 is not a divisor of 497)
497 / 497 = 1 (the remainder is 0, so 497 is a divisor of 497)

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 497 (i.e. 22.293496809608). 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 $\frac{N}{d} = D$ )

497 / 1 = 497 (the remainder is 0, so 1 and 497 are divisors of 497)
497 / 2 = 248.5 (the remainder is 1, so 2 is not a divisor of 497)
497 / 3 = 165.66666666667 (the remainder is 2, so 3 is not a divisor of 497)
...
497 / 21 = 23.666666666667 (the remainder is 14, so 21 is not a divisor of 497)
497 / 22 = 22.590909090909 (the remainder is 13, so 22 is not a divisor of 497)