What are the divisors of 1810?

1, 2, 5, 10, 181, 362, 905, 1810

4 even divisors

2, 10, 362, 1810

4 odd divisors

1, 5, 181, 905

How to compute the divisors of 1810?

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 1810 by each of the numbers from 1 to 1810 to determine which ones have a remainder equal to 0.

Remainder = N ( M × N M )

(where N M is the integer part of the quotient)

  • 1810 / 1 = 1810 (the remainder is 0, so 1 is a divisor of 1810)
  • 1810 / 2 = 905 (the remainder is 0, so 2 is a divisor of 1810)
  • 1810 / 3 = 603.33333333333 (the remainder is 1, so 3 is not a divisor of 1810)
  • ...
  • 1810 / 1809 = 1.0005527915976 (the remainder is 1, so 1809 is not a divisor of 1810)
  • 1810 / 1810 = 1 (the remainder is 0, so 1810 is a divisor of 1810)

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 1810 (i.e. 42.544094772365). 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 × d = N

(thus, if N D = d , then N d = D )

  • 1810 / 1 = 1810 (the remainder is 0, so 1 and 1810 are divisors of 1810)
  • 1810 / 2 = 905 (the remainder is 0, so 2 and 905 are divisors of 1810)
  • 1810 / 3 = 603.33333333333 (the remainder is 1, so 3 is not a divisor of 1810)
  • ...
  • 1810 / 41 = 44.146341463415 (the remainder is 6, so 41 is not a divisor of 1810)
  • 1810 / 42 = 43.095238095238 (the remainder is 4, so 42 is not a divisor of 1810)