What are the divisors of 1697?

1, 1697

There is a total of 2 positive divisors.
The sum of these divisors is 1698.
The arithmetic mean is 849.

2 odd divisors

1, 1697

How to compute the divisors of 1697?

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 1697 by each of the numbers from 1 to 1697 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)

1697 / 1 = 1697 (the remainder is 0, so 1 is a divisor of 1697)
1697 / 2 = 848.5 (the remainder is 1, so 2 is not a divisor of 1697)
1697 / 3 = 565.66666666667 (the remainder is 2, so 3 is not a divisor of 1697)
...
1697 / 1696 = 1.0005896226415 (the remainder is 1, so 1696 is not a divisor of 1697)
1697 / 1697 = 1 (the remainder is 0, so 1697 is a divisor of 1697)

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 1697 (i.e. 41.194659848092). 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 $\frac{N}{d} = D$ )

1697 / 1 = 1697 (the remainder is 0, so 1 and 1697 are divisors of 1697)
1697 / 2 = 848.5 (the remainder is 1, so 2 is not a divisor of 1697)
1697 / 3 = 565.66666666667 (the remainder is 2, so 3 is not a divisor of 1697)
...
1697 / 40 = 42.425 (the remainder is 17, so 40 is not a divisor of 1697)
1697 / 41 = 41.390243902439 (the remainder is 16, so 41 is not a divisor of 1697)