Numbers that are not divisible by any number in the range [2, 10]

22 Jun, 2022
Given an integer N, The task is to find the count of all the numbers from 1 to N which are not divisible by any number in the range [2, 10].
Examples: 
Input: N = 12 Output: 2 1, 11 are the only numbers in range [1, 12] which are not divisible by any number from 2 to 10
Input: N = 20 Output: 5 
Approach: Total numbers from 1 to n which are not divisible by any number from 2 to 10 are equal to n minus the numbers which are divisible by some numbers from 2 to 10.
The set of numbers that are divisible by some numbers from 2 to 10 can be found as union of the set of numbers from 1 to n divisible by 2, the set of numbers divisible by 3, and so on till 10.
Note that sets of numbers divisible by 4 or 6 or 8 are subsets of the set of numbers divisible by 2, and sets of numbers divisible by 6 or 9 are subsets of the set of numbers divisible by 3. So there is no need to unite 9 sets, it is enough to unite sets for 2, 3, 5, and 7 only.
The size of the set of numbers from 1 to n divisible by 2, 3, 5, and 7 can be calculated using an inclusion-exclusion principle that says that the size of every single set should be added, the size of pairwise intersections should be subtracted, the size of all intersections of three sets should be added and so on.
The size of the set of numbers from 1 to n divisible by 2 is equal to ⌊n / 2⌋, the size of the set of numbers from 1 to n divisible by 2 and 3 is equal to ⌊n / (2 * 3)⌋ and so on.
So, the formula is n – ⌊n / 2⌋ – ⌊n / 3⌋ – ⌊n / 5⌋ – ⌊n / 7⌋ + ⌊n / (2 * 3)] + ⌊n / (2 * 5)] + ⌊n / (2 * 7)] + ⌊n / (3 * 5)] + ⌊n / (3 * 7)] + ⌊n / (5 * 7)] – ⌊n / (2 * 3 * 5)] – ⌊n / (2 * 3 * 7)] – ⌊n / (2 * 5 * 7)] – ⌊n / (3 * 5 * 7)]+ ⌊n / (2 * 3 * 5 * 7)]
Below is the implementation of the above approach:  
C++
Java
Python3
C#
PHP
Javascript
// C++ implementation of the approach#include <bits/stdc++.h>using namespace std; // Function to return the count of numbers// from 1 to N which are not divisible by// any number in the range [2, 10]int countNumbers(int n){    return n - n / 2 - n / 3 - n / 5 - n / 7           + n / 6 + n / 10 + n / 14 + n / 15 + n / 21 + n / 35           - n / 30 - n / 42 - n / 70 - n / 105 + n / 210;} // Driver codeint main(){    int n = 20;    cout << countNumbers(n);     return 0;}
// Java implementation of the approachclass GFG{     // Function to return the count of numbers// from 1 to N which are not divisible by// any number in the range [2, 10]static int countNumbers(int n){    return n - n / 2 - n / 3 - n / 5 - n / 7        + n / 6 + n / 10 + n / 14 + n / 15 + n / 21 + n / 35        - n / 30 - n / 42 - n / 70 - n / 105 + n / 210;} // Driver codepublic static void main (String[] args){    int n = 20;    System.out.println(countNumbers(n));}} // This code is contributed by mits
# Python3 implementation of the approach # Function to return the count of numbers# from 1 to N which are not divisible by# any number in the range [2, 10]def countNumbers(n):    return (n - n // 2 - n // 3 - n // 5 - n // 7 +             n // 6 + n // 10 + n // 14 + n // 15 +             n // 21 + n // 35 - n // 30 - n // 42 -             n // 70 - n // 105 + n // 210) # Driver codeif __name__ == '__main__':    n = 20    print(countNumbers(n)) # This code contributed by Rajput-Ji
// C# implementation of the approachusing System; class GFG{     // Function to return the count of numbers// from 1 to N which are not divisible by// any number in the range [2, 10]static int countNumbers(int n){    return n - n / 2 - n / 3 - n / 5 - n / 7        + n / 6 + n / 10 + n / 14 + n / 15 + n / 21 + n / 35        - n / 30 - n / 42 - n / 70 - n / 105 + n / 210;} // Driver codestatic void Main(){    int n = 20;    Console.WriteLine(countNumbers(n));}} // This code is contributed by mits
<?php// PHP implementation of the approach // Function to return the count of numbers// from 1 to N which are not divisible by// any number in the range [2, 10]function countNumbers($n){    return (int)($n - $n / 2) - (int)($n / 3 ) -           (int)($n / 5 ) - (int)($n / 7) +           (int)($n / 6 ) + (int)($n / 10) +           (int)($n / 14) + (int)($n / 15) +           (int)($n / 21) + (int)($n / 35) -           (int)($n / 30) - (int)($n / 42) -           (int)($n / 70) - (int)($n / 105) +           (int)($n / 210);} // Driver code$n = 20;echo(countNumbers($n)); // This code is contributed by Code_Mech.?>
<script> // Javascript implementation of the approach // Function to return the count of numbers// from 1 to N which are not divisible by// any number in the range [2, 10]function countNumbers(n){    return n - parseInt(n / 2, 10) - parseInt(n / 3, 10) -               parseInt(n / 5, 10) - parseInt(n / 7, 10) +               parseInt(n / 6, 10) + parseInt(n / 10, 10) +               parseInt(n / 14, 10) + parseInt(n / 15, 10) +               parseInt(n / 21, 10) + parseInt(n / 35, 10) -               parseInt(n / 30, 10) - parseInt(n / 42, 10) -               parseInt(n / 70, 10) - parseInt(n / 105, 10) +               parseInt(n / 210, 10);} // Driver codelet n = 20; document.write(countNumbers(n)); // This code is contributed by mukesh07 </script>
5
 
Time Complexity: O(1)Auxiliary Space: O(1)
Mithun Kumar
rrlinus
Rajput-Ji
Code_Mech
mukesh07
subham348
divisibility
Competitive Programming
Mathematical
Mathematical
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