What are the divisors of 1523?

1, 1523

There is a total of 2 positive divisors.
The sum of these divisors is 1524.
The arithmetic mean is 762.

2 odd divisors

1, 1523

How to compute the divisors of 1523?

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 1523 by each of the numbers from 1 to 1523 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)

1523 / 1 = 1523 (the remainder is 0, so 1 is a divisor of 1523)
1523 / 2 = 761.5 (the remainder is 1, so 2 is not a divisor of 1523)
1523 / 3 = 507.66666666667 (the remainder is 2, so 3 is not a divisor of 1523)
...
1523 / 1522 = 1.0006570302234 (the remainder is 1, so 1522 is not a divisor of 1523)
1523 / 1523 = 1 (the remainder is 0, so 1523 is a divisor of 1523)

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 1523 (i.e. 39.025632602176). 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 $\frac{N}{d} = D$ )

1523 / 1 = 1523 (the remainder is 0, so 1 and 1523 are divisors of 1523)
1523 / 2 = 761.5 (the remainder is 1, so 2 is not a divisor of 1523)
1523 / 3 = 507.66666666667 (the remainder is 2, so 3 is not a divisor of 1523)
...
1523 / 38 = 40.078947368421 (the remainder is 3, so 38 is not a divisor of 1523)
1523 / 39 = 39.051282051282 (the remainder is 2, so 39 is not a divisor of 1523)