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Program to find nth term of a sequence which are divisible by a, b, c in Python

Suppose we have four numbers n, a, b, and c. We have to find the nth (0 indexed) term of the sorted sequence of numbers divisible by a, b or c.

So, if the input is like n = 8 a = 3 b = 7 c = 9, then the output will be 18, as The first 9 terms of the sequence are [1, 3, 6, 7, 9, 12, 14, 15, 18].

To solve this, we will follow these steps −

  • if minimum of a, b, c is same as 1, then
    • return n
  • ab := lcm(a, b), bc := lcm(b, c), ca := lcm(a, c)
  • abc := lcm(ab, c)
  • left := 1, right := 10^9
  • while left <= right, do
    • mid :=(left + right) / 2
    • na := quotient of mid / a
    • nb := quotient of mid / b
    • nc := quotient of mid / c
    • nab := quotient of mid / ab
    • nbc := quotient of mid / bc
    • nca := quotient of mid / ca
    • nabc := quotient of mid / abc
    • numterms := na + nb + nc - nab - nbc - nca + nabc
    • if numterms > n, then
      • right := mid - 1
    • otherwise when numterms < n, then
      • left := mid + 1
    • otherwise,
      • return mid - minimum of (mid mod a, mid mod b, mid mod c)
  • return -1

Example (Python)

Let us see the following implementation to get better understanding −

import math
def lcm(a, b):
   return (a * b) // math.gcd(a, b)
class Solution:
   def solve(self, n, a, b, c):
      if min(a, b, c) == 1:
         return n
      ab, bc, ca = lcm(a, b), lcm(b, c), lcm(a, c)
      abc = lcm(ab, c)
      left, right = 1, 10 ** 9
      while left <= right:
         mid = (left + right) // 2
         na = mid // a
         nb = mid // b
         nc = mid // c
         nab = mid // ab
         nbc = mid // bc
         nca = mid // ca
         nabc = mid // abc
         numterms = na + nb + nc - nab - nbc - nca + nabc
         if numterms > n:
            right = mid - 1
         elif numterms < n:
            left = mid + 1
         else:
            return mid - min(mid % a, mid % b, mid % c)
      return -1
ob = Solution()
n = 8
a = 3
b = 7
c = 9
print(ob.solve(n, a, b, c))

Input

8, 3, 7, 9

Output

18