What are the divisors of 343?

1, 7, 49, 343

There is a total of 4 positive divisors.
The sum of these divisors is 400.
The arithmetic mean is 100.

4 odd divisors

1, 7, 49, 343

How to compute the divisors of 343?

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 343 by each of the numbers from 1 to 343 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)

343 / 1 = 343 (the remainder is 0, so 1 is a divisor of 343)
343 / 2 = 171.5 (the remainder is 1, so 2 is not a divisor of 343)
343 / 3 = 114.33333333333 (the remainder is 1, so 3 is not a divisor of 343)
...
343 / 342 = 1.0029239766082 (the remainder is 1, so 342 is not a divisor of 343)
343 / 343 = 1 (the remainder is 0, so 343 is a divisor of 343)

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 343 (i.e. 18.520259177452). 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 $\frac{N}{d} = D$ )

343 / 1 = 343 (the remainder is 0, so 1 and 343 are divisors of 343)
343 / 2 = 171.5 (the remainder is 1, so 2 is not a divisor of 343)
343 / 3 = 114.33333333333 (the remainder is 1, so 3 is not a divisor of 343)
...
343 / 17 = 20.176470588235 (the remainder is 3, so 17 is not a divisor of 343)
343 / 18 = 19.055555555556 (the remainder is 1, so 18 is not a divisor of 343)