What are the divisors of 763?

1, 7, 109, 763

There is a total of 4 positive divisors.
The sum of these divisors is 880.
The arithmetic mean is 220.

4 odd divisors

1, 7, 109, 763

How to compute the divisors of 763?

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 763 by each of the numbers from 1 to 763 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)

763 / 1 = 763 (the remainder is 0, so 1 is a divisor of 763)
763 / 2 = 381.5 (the remainder is 1, so 2 is not a divisor of 763)
763 / 3 = 254.33333333333 (the remainder is 1, so 3 is not a divisor of 763)
...
763 / 762 = 1.001312335958 (the remainder is 1, so 762 is not a divisor of 763)
763 / 763 = 1 (the remainder is 0, so 763 is a divisor of 763)

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 763 (i.e. 27.622454633866). 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 $\frac{N}{d} = D$ )

763 / 1 = 763 (the remainder is 0, so 1 and 763 are divisors of 763)
763 / 2 = 381.5 (the remainder is 1, so 2 is not a divisor of 763)
763 / 3 = 254.33333333333 (the remainder is 1, so 3 is not a divisor of 763)
...
763 / 26 = 29.346153846154 (the remainder is 9, so 26 is not a divisor of 763)
763 / 27 = 28.259259259259 (the remainder is 7, so 27 is not a divisor of 763)