What are the divisors of 1500?

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 125, 150, 250, 300, 375, 500, 750, 1500

16 even divisors

2, 4, 6, 10, 12, 20, 30, 50, 60, 100, 150, 250, 300, 500, 750, 1500

8 odd divisors

1, 3, 5, 15, 25, 75, 125, 375

How to compute the divisors of 1500?

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 1500 by each of the numbers from 1 to 1500 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)

  • 1500 / 1 = 1500 (the remainder is 0, so 1 is a divisor of 1500)
  • 1500 / 2 = 750 (the remainder is 0, so 2 is a divisor of 1500)
  • 1500 / 3 = 500 (the remainder is 0, so 3 is a divisor of 1500)
  • ...
  • 1500 / 1499 = 1.0006671114076 (the remainder is 1, so 1499 is not a divisor of 1500)
  • 1500 / 1500 = 1 (the remainder is 0, so 1500 is a divisor of 1500)

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 1500 (i.e. 38.729833462074). 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 )

  • 1500 / 1 = 1500 (the remainder is 0, so 1 and 1500 are divisors of 1500)
  • 1500 / 2 = 750 (the remainder is 0, so 2 and 750 are divisors of 1500)
  • 1500 / 3 = 500 (the remainder is 0, so 3 and 500 are divisors of 1500)
  • ...
  • 1500 / 37 = 40.540540540541 (the remainder is 20, so 37 is not a divisor of 1500)
  • 1500 / 38 = 39.473684210526 (the remainder is 18, so 38 is not a divisor of 1500)