What are the divisors of 697?

1, 17, 41, 697

4 odd divisors

1, 17, 41, 697

How to compute the divisors of 697?

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 697 by each of the numbers from 1 to 697 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)

  • 697 / 1 = 697 (the remainder is 0, so 1 is a divisor of 697)
  • 697 / 2 = 348.5 (the remainder is 1, so 2 is not a divisor of 697)
  • 697 / 3 = 232.33333333333 (the remainder is 1, so 3 is not a divisor of 697)
  • ...
  • 697 / 696 = 1.0014367816092 (the remainder is 1, so 696 is not a divisor of 697)
  • 697 / 697 = 1 (the remainder is 0, so 697 is a divisor of 697)

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 697 (i.e. 26.400757564888). 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 )

  • 697 / 1 = 697 (the remainder is 0, so 1 and 697 are divisors of 697)
  • 697 / 2 = 348.5 (the remainder is 1, so 2 is not a divisor of 697)
  • 697 / 3 = 232.33333333333 (the remainder is 1, so 3 is not a divisor of 697)
  • ...
  • 697 / 25 = 27.88 (the remainder is 22, so 25 is not a divisor of 697)
  • 697 / 26 = 26.807692307692 (the remainder is 21, so 26 is not a divisor of 697)