What are the divisors of 1410?

1, 2, 3, 5, 6, 10, 15, 30, 47, 94, 141, 235, 282, 470, 705, 1410

There is a total of 16 positive divisors.
The sum of these divisors is 3456.
The arithmetic mean is 216.

8 even divisors

2, 6, 10, 30, 94, 282, 470, 1410

8 odd divisors

1, 3, 5, 15, 47, 141, 235, 705

How to compute the divisors of 1410?

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 1410 by each of the numbers from 1 to 1410 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)

1410 / 1 = 1410 (the remainder is 0, so 1 is a divisor of 1410)
1410 / 2 = 705 (the remainder is 0, so 2 is a divisor of 1410)
1410 / 3 = 470 (the remainder is 0, so 3 is a divisor of 1410)
...
1410 / 1409 = 1.0007097232079 (the remainder is 1, so 1409 is not a divisor of 1410)
1410 / 1410 = 1 (the remainder is 0, so 1410 is a divisor of 1410)

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 1410 (i.e. 37.549966711037). 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 $\frac{N}{d} = D$ )

1410 / 1 = 1410 (the remainder is 0, so 1 and 1410 are divisors of 1410)
1410 / 2 = 705 (the remainder is 0, so 2 and 705 are divisors of 1410)
1410 / 3 = 470 (the remainder is 0, so 3 and 470 are divisors of 1410)
...
1410 / 36 = 39.166666666667 (the remainder is 6, so 36 is not a divisor of 1410)
1410 / 37 = 38.108108108108 (the remainder is 4, so 37 is not a divisor of 1410)