What are the divisors of 1070?

1, 2, 5, 10, 107, 214, 535, 1070

There is a total of 8 positive divisors.
The sum of these divisors is 1944.
The arithmetic mean is 243.

4 even divisors

2, 10, 214, 1070

4 odd divisors

1, 5, 107, 535

How to compute the divisors of 1070?

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 1070 by each of the numbers from 1 to 1070 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)

1070 / 1 = 1070 (the remainder is 0, so 1 is a divisor of 1070)
1070 / 2 = 535 (the remainder is 0, so 2 is a divisor of 1070)
1070 / 3 = 356.66666666667 (the remainder is 2, so 3 is not a divisor of 1070)
...
1070 / 1069 = 1.000935453695 (the remainder is 1, so 1069 is not a divisor of 1070)
1070 / 1070 = 1 (the remainder is 0, so 1070 is a divisor of 1070)

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 1070 (i.e. 32.710854467592). 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 $\frac{N}{d} = D$ )

1070 / 1 = 1070 (the remainder is 0, so 1 and 1070 are divisors of 1070)
1070 / 2 = 535 (the remainder is 0, so 2 and 535 are divisors of 1070)
1070 / 3 = 356.66666666667 (the remainder is 2, so 3 is not a divisor of 1070)
...
1070 / 31 = 34.516129032258 (the remainder is 16, so 31 is not a divisor of 1070)
1070 / 32 = 33.4375 (the remainder is 14, so 32 is not a divisor of 1070)