What are the divisors of 1742?

1, 2, 13, 26, 67, 134, 871, 1742

There is a total of 8 positive divisors.
The sum of these divisors is 2856.
The arithmetic mean is 357.

4 even divisors

2, 26, 134, 1742

4 odd divisors

1, 13, 67, 871

How to compute the divisors of 1742?

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 1742 by each of the numbers from 1 to 1742 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)

1742 / 1 = 1742 (the remainder is 0, so 1 is a divisor of 1742)
1742 / 2 = 871 (the remainder is 0, so 2 is a divisor of 1742)
1742 / 3 = 580.66666666667 (the remainder is 2, so 3 is not a divisor of 1742)
...
1742 / 1741 = 1.0005743825388 (the remainder is 1, so 1741 is not a divisor of 1742)
1742 / 1742 = 1 (the remainder is 0, so 1742 is a divisor of 1742)

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 1742 (i.e. 41.737273509418). 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 $\frac{N}{d} = D$ )

1742 / 1 = 1742 (the remainder is 0, so 1 and 1742 are divisors of 1742)
1742 / 2 = 871 (the remainder is 0, so 2 and 871 are divisors of 1742)
1742 / 3 = 580.66666666667 (the remainder is 2, so 3 is not a divisor of 1742)
...
1742 / 40 = 43.55 (the remainder is 22, so 40 is not a divisor of 1742)
1742 / 41 = 42.487804878049 (the remainder is 20, so 41 is not a divisor of 1742)