What are the divisors of 1610?

1, 2, 5, 7, 10, 14, 23, 35, 46, 70, 115, 161, 230, 322, 805, 1610

8 even divisors

2, 10, 14, 46, 70, 230, 322, 1610

8 odd divisors

1, 5, 7, 23, 35, 115, 161, 805

How to compute the divisors of 1610?

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 1610 by each of the numbers from 1 to 1610 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)

  • 1610 / 1 = 1610 (the remainder is 0, so 1 is a divisor of 1610)
  • 1610 / 2 = 805 (the remainder is 0, so 2 is a divisor of 1610)
  • 1610 / 3 = 536.66666666667 (the remainder is 2, so 3 is not a divisor of 1610)
  • ...
  • 1610 / 1609 = 1.0006215040398 (the remainder is 1, so 1609 is not a divisor of 1610)
  • 1610 / 1610 = 1 (the remainder is 0, so 1610 is a divisor of 1610)

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 1610 (i.e. 40.124805295478). 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 )

  • 1610 / 1 = 1610 (the remainder is 0, so 1 and 1610 are divisors of 1610)
  • 1610 / 2 = 805 (the remainder is 0, so 2 and 805 are divisors of 1610)
  • 1610 / 3 = 536.66666666667 (the remainder is 2, so 3 is not a divisor of 1610)
  • ...
  • 1610 / 39 = 41.282051282051 (the remainder is 11, so 39 is not a divisor of 1610)
  • 1610 / 40 = 40.25 (the remainder is 10, so 40 is not a divisor of 1610)