What are the divisors of 1400?

1, 2, 4, 5, 7, 8, 10, 14, 20, 25, 28, 35, 40, 50, 56, 70, 100, 140, 175, 200, 280, 350, 700, 1400

There is a total of 24 positive divisors.
The sum of these divisors is 3720.
The arithmetic mean is 155.

18 even divisors

2, 4, 8, 10, 14, 20, 28, 40, 50, 56, 70, 100, 140, 200, 280, 350, 700, 1400

6 odd divisors

1, 5, 7, 25, 35, 175

How to compute the divisors of 1400?

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 1400 by each of the numbers from 1 to 1400 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)

1400 / 1 = 1400 (the remainder is 0, so 1 is a divisor of 1400)
1400 / 2 = 700 (the remainder is 0, so 2 is a divisor of 1400)
1400 / 3 = 466.66666666667 (the remainder is 2, so 3 is not a divisor of 1400)
...
1400 / 1399 = 1.0007147962831 (the remainder is 1, so 1399 is not a divisor of 1400)
1400 / 1400 = 1 (the remainder is 0, so 1400 is a divisor of 1400)

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 1400 (i.e. 37.416573867739). 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 $\frac{N}{d} = D$ )

1400 / 1 = 1400 (the remainder is 0, so 1 and 1400 are divisors of 1400)
1400 / 2 = 700 (the remainder is 0, so 2 and 700 are divisors of 1400)
1400 / 3 = 466.66666666667 (the remainder is 2, so 3 is not a divisor of 1400)
...
1400 / 36 = 38.888888888889 (the remainder is 32, so 36 is not a divisor of 1400)
1400 / 37 = 37.837837837838 (the remainder is 31, so 37 is not a divisor of 1400)