What are the divisors of 1110?

1, 2, 3, 5, 6, 10, 15, 30, 37, 74, 111, 185, 222, 370, 555, 1110

There is a total of 16 positive divisors.
The sum of these divisors is 2736.
The arithmetic mean is 171.

8 even divisors

2, 6, 10, 30, 74, 222, 370, 1110

8 odd divisors

1, 3, 5, 15, 37, 111, 185, 555

How to compute the divisors of 1110?

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 1110 by each of the numbers from 1 to 1110 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)

1110 / 1 = 1110 (the remainder is 0, so 1 is a divisor of 1110)
1110 / 2 = 555 (the remainder is 0, so 2 is a divisor of 1110)
1110 / 3 = 370 (the remainder is 0, so 3 is a divisor of 1110)
...
1110 / 1109 = 1.0009017132552 (the remainder is 1, so 1109 is not a divisor of 1110)
1110 / 1110 = 1 (the remainder is 0, so 1110 is a divisor of 1110)

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 1110 (i.e. 33.316662497915). 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 $\frac{N}{d} = D$ )

1110 / 1 = 1110 (the remainder is 0, so 1 and 1110 are divisors of 1110)
1110 / 2 = 555 (the remainder is 0, so 2 and 555 are divisors of 1110)
1110 / 3 = 370 (the remainder is 0, so 3 and 370 are divisors of 1110)
...
1110 / 32 = 34.6875 (the remainder is 22, so 32 is not a divisor of 1110)
1110 / 33 = 33.636363636364 (the remainder is 21, so 33 is not a divisor of 1110)