What are the divisors of 1407?

1, 3, 7, 21, 67, 201, 469, 1407

There is a total of 8 positive divisors.
The sum of these divisors is 2176.
The arithmetic mean is 272.

8 odd divisors

1, 3, 7, 21, 67, 201, 469, 1407

How to compute the divisors of 1407?

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 1407 by each of the numbers from 1 to 1407 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)

1407 / 1 = 1407 (the remainder is 0, so 1 is a divisor of 1407)
1407 / 2 = 703.5 (the remainder is 1, so 2 is not a divisor of 1407)
1407 / 3 = 469 (the remainder is 0, so 3 is a divisor of 1407)
...
1407 / 1406 = 1.0007112375533 (the remainder is 1, so 1406 is not a divisor of 1407)
1407 / 1407 = 1 (the remainder is 0, so 1407 is a divisor of 1407)

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 1407 (i.e. 37.509998667022). 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 $\frac{N}{d} = D$ )

1407 / 1 = 1407 (the remainder is 0, so 1 and 1407 are divisors of 1407)
1407 / 2 = 703.5 (the remainder is 1, so 2 is not a divisor of 1407)
1407 / 3 = 469 (the remainder is 0, so 3 and 469 are divisors of 1407)
...
1407 / 36 = 39.083333333333 (the remainder is 3, so 36 is not a divisor of 1407)
1407 / 37 = 38.027027027027 (the remainder is 1, so 37 is not a divisor of 1407)