What are the divisors of 1420?

1, 2, 4, 5, 10, 20, 71, 142, 284, 355, 710, 1420

There is a total of 12 positive divisors.
The sum of these divisors is 3024.
The arithmetic mean is 252.

8 even divisors

2, 4, 10, 20, 142, 284, 710, 1420

4 odd divisors

1, 5, 71, 355

How to compute the divisors of 1420?

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 1420 by each of the numbers from 1 to 1420 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)

1420 / 1 = 1420 (the remainder is 0, so 1 is a divisor of 1420)
1420 / 2 = 710 (the remainder is 0, so 2 is a divisor of 1420)
1420 / 3 = 473.33333333333 (the remainder is 1, so 3 is not a divisor of 1420)
...
1420 / 1419 = 1.000704721635 (the remainder is 1, so 1419 is not a divisor of 1420)
1420 / 1420 = 1 (the remainder is 0, so 1420 is a divisor of 1420)

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 1420 (i.e. 37.682887362834). 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 $\frac{N}{d} = D$ )

1420 / 1 = 1420 (the remainder is 0, so 1 and 1420 are divisors of 1420)
1420 / 2 = 710 (the remainder is 0, so 2 and 710 are divisors of 1420)
1420 / 3 = 473.33333333333 (the remainder is 1, so 3 is not a divisor of 1420)
...
1420 / 36 = 39.444444444444 (the remainder is 16, so 36 is not a divisor of 1420)
1420 / 37 = 38.378378378378 (the remainder is 14, so 37 is not a divisor of 1420)