What are the divisors of 1412?

1, 2, 4, 353, 706, 1412

There is a total of 6 positive divisors.
The sum of these divisors is 2478.
The arithmetic mean is 413.

4 even divisors

2, 4, 706, 1412

2 odd divisors

1, 353

How to compute the divisors of 1412?

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 1412 by each of the numbers from 1 to 1412 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)

1412 / 1 = 1412 (the remainder is 0, so 1 is a divisor of 1412)
1412 / 2 = 706 (the remainder is 0, so 2 is a divisor of 1412)
1412 / 3 = 470.66666666667 (the remainder is 2, so 3 is not a divisor of 1412)
...
1412 / 1411 = 1.0007087172218 (the remainder is 1, so 1411 is not a divisor of 1412)
1412 / 1412 = 1 (the remainder is 0, so 1412 is a divisor of 1412)

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 1412 (i.e. 37.576588456112). 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 $\frac{N}{d} = D$ )

1412 / 1 = 1412 (the remainder is 0, so 1 and 1412 are divisors of 1412)
1412 / 2 = 706 (the remainder is 0, so 2 and 706 are divisors of 1412)
1412 / 3 = 470.66666666667 (the remainder is 2, so 3 is not a divisor of 1412)
...
1412 / 36 = 39.222222222222 (the remainder is 8, so 36 is not a divisor of 1412)
1412 / 37 = 38.162162162162 (the remainder is 6, so 37 is not a divisor of 1412)