What are the divisors of 1694?

1, 2, 7, 11, 14, 22, 77, 121, 154, 242, 847, 1694

There is a total of 12 positive divisors.
The sum of these divisors is 3192.
The arithmetic mean is 266.

6 even divisors

2, 14, 22, 154, 242, 1694

6 odd divisors

1, 7, 11, 77, 121, 847

How to compute the divisors of 1694?

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 1694 by each of the numbers from 1 to 1694 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)

1694 / 1 = 1694 (the remainder is 0, so 1 is a divisor of 1694)
1694 / 2 = 847 (the remainder is 0, so 2 is a divisor of 1694)
1694 / 3 = 564.66666666667 (the remainder is 2, so 3 is not a divisor of 1694)
...
1694 / 1693 = 1.0005906674542 (the remainder is 1, so 1693 is not a divisor of 1694)
1694 / 1694 = 1 (the remainder is 0, so 1694 is a divisor of 1694)

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 1694 (i.e. 41.158231254513). 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 $\frac{N}{d} = D$ )

1694 / 1 = 1694 (the remainder is 0, so 1 and 1694 are divisors of 1694)
1694 / 2 = 847 (the remainder is 0, so 2 and 847 are divisors of 1694)
1694 / 3 = 564.66666666667 (the remainder is 2, so 3 is not a divisor of 1694)
...
1694 / 40 = 42.35 (the remainder is 14, so 40 is not a divisor of 1694)
1694 / 41 = 41.317073170732 (the remainder is 13, so 41 is not a divisor of 1694)