What are the divisors of 698?

1, 2, 349, 698

There is a total of 4 positive divisors.
The sum of these divisors is 1050.
The arithmetic mean is 262.5.

2 even divisors

2, 698

2 odd divisors

1, 349

How to compute the divisors of 698?

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 698 by each of the numbers from 1 to 698 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)

698 / 1 = 698 (the remainder is 0, so 1 is a divisor of 698)
698 / 2 = 349 (the remainder is 0, so 2 is a divisor of 698)
698 / 3 = 232.66666666667 (the remainder is 2, so 3 is not a divisor of 698)
...
698 / 697 = 1.0014347202296 (the remainder is 1, so 697 is not a divisor of 698)
698 / 698 = 1 (the remainder is 0, so 698 is a divisor of 698)

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 698 (i.e. 26.419689627246). 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 $\frac{N}{d} = D$ )

698 / 1 = 698 (the remainder is 0, so 1 and 698 are divisors of 698)
698 / 2 = 349 (the remainder is 0, so 2 and 349 are divisors of 698)
698 / 3 = 232.66666666667 (the remainder is 2, so 3 is not a divisor of 698)
...
698 / 25 = 27.92 (the remainder is 23, so 25 is not a divisor of 698)
698 / 26 = 26.846153846154 (the remainder is 22, so 26 is not a divisor of 698)