What are the divisors of 234?

1, 2, 3, 6, 9, 13, 18, 26, 39, 78, 117, 234

6 even divisors

2, 6, 18, 26, 78, 234

6 odd divisors

1, 3, 9, 13, 39, 117

How to compute the divisors of 234?

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 234 by each of the numbers from 1 to 234 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)

  • 234 / 1 = 234 (the remainder is 0, so 1 is a divisor of 234)
  • 234 / 2 = 117 (the remainder is 0, so 2 is a divisor of 234)
  • 234 / 3 = 78 (the remainder is 0, so 3 is a divisor of 234)
  • ...
  • 234 / 233 = 1.0042918454936 (the remainder is 1, so 233 is not a divisor of 234)
  • 234 / 234 = 1 (the remainder is 0, so 234 is a divisor of 234)

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 234 (i.e. 15.297058540778). 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 )

  • 234 / 1 = 234 (the remainder is 0, so 1 and 234 are divisors of 234)
  • 234 / 2 = 117 (the remainder is 0, so 2 and 117 are divisors of 234)
  • 234 / 3 = 78 (the remainder is 0, so 3 and 78 are divisors of 234)
  • ...
  • 234 / 14 = 16.714285714286 (the remainder is 10, so 14 is not a divisor of 234)
  • 234 / 15 = 15.6 (the remainder is 9, so 15 is not a divisor of 234)