What are the divisors of 973?

1, 7, 139, 973

There is a total of 4 positive divisors.
The sum of these divisors is 1120.
The arithmetic mean is 280.

4 odd divisors

1, 7, 139, 973

How to compute the divisors of 973?

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 973 by each of the numbers from 1 to 973 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)

973 / 1 = 973 (the remainder is 0, so 1 is a divisor of 973)
973 / 2 = 486.5 (the remainder is 1, so 2 is not a divisor of 973)
973 / 3 = 324.33333333333 (the remainder is 1, so 3 is not a divisor of 973)
...
973 / 972 = 1.0010288065844 (the remainder is 1, so 972 is not a divisor of 973)
973 / 973 = 1 (the remainder is 0, so 973 is a divisor of 973)

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 973 (i.e. 31.192947920964). 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 $\frac{N}{d} = D$ )

973 / 1 = 973 (the remainder is 0, so 1 and 973 are divisors of 973)
973 / 2 = 486.5 (the remainder is 1, so 2 is not a divisor of 973)
973 / 3 = 324.33333333333 (the remainder is 1, so 3 is not a divisor of 973)
...
973 / 30 = 32.433333333333 (the remainder is 13, so 30 is not a divisor of 973)
973 / 31 = 31.387096774194 (the remainder is 12, so 31 is not a divisor of 973)