What are the divisors of 1521?

1, 3, 9, 13, 39, 117, 169, 507, 1521

9 odd divisors

1, 3, 9, 13, 39, 117, 169, 507, 1521

How to compute the divisors of 1521?

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 1521 by each of the numbers from 1 to 1521 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)

  • 1521 / 1 = 1521 (the remainder is 0, so 1 is a divisor of 1521)
  • 1521 / 2 = 760.5 (the remainder is 1, so 2 is not a divisor of 1521)
  • 1521 / 3 = 507 (the remainder is 0, so 3 is a divisor of 1521)
  • ...
  • 1521 / 1520 = 1.0006578947368 (the remainder is 1, so 1520 is not a divisor of 1521)
  • 1521 / 1521 = 1 (the remainder is 0, so 1521 is a divisor of 1521)

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 1521 (i.e. 39). 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 )

  • 1521 / 1 = 1521 (the remainder is 0, so 1 and 1521 are divisors of 1521)
  • 1521 / 2 = 760.5 (the remainder is 1, so 2 is not a divisor of 1521)
  • 1521 / 3 = 507 (the remainder is 0, so 3 and 507 are divisors of 1521)
  • ...
  • 1521 / 38 = 40.026315789474 (the remainder is 1, so 38 is not a divisor of 1521)
  • 1521 / 39 = 39 (the remainder is 0, so 39 and 39 are divisors of 1521)