What are the divisors of 1099?

1, 7, 157, 1099

There is a total of 4 positive divisors.
The sum of these divisors is 1264.
The arithmetic mean is 316.

4 odd divisors

1, 7, 157, 1099

How to compute the divisors of 1099?

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 1099 by each of the numbers from 1 to 1099 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)

1099 / 1 = 1099 (the remainder is 0, so 1 is a divisor of 1099)
1099 / 2 = 549.5 (the remainder is 1, so 2 is not a divisor of 1099)
1099 / 3 = 366.33333333333 (the remainder is 1, so 3 is not a divisor of 1099)
...
1099 / 1098 = 1.0009107468124 (the remainder is 1, so 1098 is not a divisor of 1099)
1099 / 1099 = 1 (the remainder is 0, so 1099 is a divisor of 1099)

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 1099 (i.e. 33.151168908502). 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 $\frac{N}{d} = D$ )

1099 / 1 = 1099 (the remainder is 0, so 1 and 1099 are divisors of 1099)
1099 / 2 = 549.5 (the remainder is 1, so 2 is not a divisor of 1099)
1099 / 3 = 366.33333333333 (the remainder is 1, so 3 is not a divisor of 1099)
...
1099 / 32 = 34.34375 (the remainder is 11, so 32 is not a divisor of 1099)
1099 / 33 = 33.30303030303 (the remainder is 10, so 33 is not a divisor of 1099)