What are the divisors of 1354?

1, 2, 677, 1354

There is a total of 4 positive divisors.
The sum of these divisors is 2034.
The arithmetic mean is 508.5.

2 even divisors

2, 1354

2 odd divisors

1, 677

How to compute the divisors of 1354?

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 1354 by each of the numbers from 1 to 1354 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)

1354 / 1 = 1354 (the remainder is 0, so 1 is a divisor of 1354)
1354 / 2 = 677 (the remainder is 0, so 2 is a divisor of 1354)
1354 / 3 = 451.33333333333 (the remainder is 1, so 3 is not a divisor of 1354)
...
1354 / 1353 = 1.0007390983001 (the remainder is 1, so 1353 is not a divisor of 1354)
1354 / 1354 = 1 (the remainder is 0, so 1354 is a divisor of 1354)

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 1354 (i.e. 36.796738985948). 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 $\frac{N}{d} = D$ )

1354 / 1 = 1354 (the remainder is 0, so 1 and 1354 are divisors of 1354)
1354 / 2 = 677 (the remainder is 0, so 2 and 677 are divisors of 1354)
1354 / 3 = 451.33333333333 (the remainder is 1, so 3 is not a divisor of 1354)
...
1354 / 35 = 38.685714285714 (the remainder is 24, so 35 is not a divisor of 1354)
1354 / 36 = 37.611111111111 (the remainder is 22, so 36 is not a divisor of 1354)