Minimum Removals from Array to Make GCD Greater in Python

Suppose we have a list of N numbers; we have to find the minimum number of removal of numbers are required so that the GCD of the remaining numbers is larger than initial GCD of N numbers.

So, if the input is like [6,9,15,30], then the output will be 2 as the initial gcd is 3, so after removing 6 and 9 we can get gcd as 15, 15 > 3.

To solve this, we will follow these steps −

• INF := 100001
• spf := a list with elements 0 to INF
• Define a function sieve()
• for i in range 4 to INF, increase by 2, do
  • spf[i] := 2
• for i in range 3 to INF, do
  • if i^2 > INF −
    • break
  • if spf[i] is same as i, then
    • for j in range 2 * i to INF, update in each step by i, do
      • if spf[j] is same as j, then
        • spf[j] := i
• Define a function calc_fact(). This will take x
• ret := a new list
• while x is not same as 1, do
  • insert spf[x] at the end of ret
  • x := x / spf[x] (get only integer part)
• return ret
• From the main method do the following −
• g := 0
• for i in range 0 to n, do
  • g := gcd(a[i], g)
• my_map := a new map
• for i in range 0 to n, do
  • a[i] := a[i] / g (get only integer part)
• for i in range 0 to n, do
  • p := calc_fact(a[i])
  • s := a new map
  • for j in range 0 to size of p, do
    • s[p[j]] := 1
  • for each i in s, do
    • my_map[i] := get(i, 0) of my_map + 1
• minimum = 10^9
• for each i in my_map, do
  • first := i
  • second := my_map[i]
  • if (n - second) <= minimum, then
    • minimum := n - second
• if minimum is not 10^9, then
  • return minimum
• otherwise,
  • return -1

Example

Let us see the following implementation to get better understanding −

from math import gcd as __gcd
INF = 100001
spf = [i for i in range(INF)]
def sieve():
   for i in range(4, INF, 2):
      spf[i] = 2
   for i in range(3, INF):
      if i**2 > INF:
         break
      if (spf[i] == i):
         for j in range(2 * i, INF, i):
            if (spf[j] == j):
               spf[j] = i
def calc_fact(x):
   ret = []
   while (x != 1):
      ret.append(spf[x])
      x = x // spf[x]
   return ret
def minRemove(a, n):
   g = 0
   for i in range(n):
      g = __gcd(a[i], g)
   my_map = dict()
   for i in range(n):
      a[i] = a[i] // g
   for i in range(n):
      p = calc_fact(a[i])
      s = dict()
      for j in range(len(p)):
         s[p[j]] = 1
      for i in s:
         my_map[i] = my_map.get(i, 0) + 1
   minimum = 10**9
   for i in my_map:
      first = i
      second = my_map[i]
      if ((n - second) <= minimum):
         minimum = n - second
   if (minimum != 10**9):
      return minimum
   else:
      return -1
a = [6, 9, 15, 30]
n = len(a)
sieve()
print(minRemove(a, n))

Input

[6, 9, 15, 30], 4

Output

2