Given a positive integer N. In one operation subtract N with its highest divisor other than N and 1. The task is to find minimum operations required to reduce N exactly to a prime number.
Examples:
Input: N = 38
Output: 1
Explanation: Highest divisor of 38 is 19, so subtract it (38 - 19) = 19. 19 is a prime number.
So, number of operations required = 1.Input: N = 69
Output: 2
Approach: This problem can be solved by using simple concepts of maths. Follow the steps below to solve the given problem.
- At first, check whether N is already prime or not.
- If N is already a prime, return 0.
- Else, Initialize a variable say count = 0 to store the number of operations required.
- Initialize a variable say i=2 and run a while loop till N!=i
- Run while loop and on each iteration subtract current value of N with its largest divisor.
- Compute the steps and increment the count by 1.
- Return the count.
Below is the implementation of the above approach:
// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
// Function check whether a number
// is prime or not
bool isPrime(int n)
{
// Corner case
if (n <= 1)
return false;
// Check from 2 to square root of n
for (int i = 2; i <= sqrt(n); i++)
if (n % i == 0)
return false;
return true;
}
// Function to minimum operations required
// to reduce the number to a prime number
int minOperation(int N)
{
// Because 1 cannot be converted to prime
if (N == 1)
return -1;
// If given number is already prime
// return 0
if (isPrime(N) == true) {
return 0;
}
// If number is not prime
else {
// Variable for total count
int count = 0;
int i = 2;
// If number is not equal to i
while (N != i) {
// If N is completely divisible by i
while (N % i == 0) {
// Temporary variable to store
// current number
int temp = N;
// Update the number by decrementing
// with highest divisor
N -= (temp / i);
// Increment count by 1
count++;
if (isPrime(N))
return count;
}
i++;
}
// Return the count
return count;
}
}
// Driver Code
int main()
{
int N = 38;
cout << minOperation(N);
return 0;
}
// Java program for the above approach
import java.util.Arrays;
class GFG {
// Function check whether a number
// is prime or not
public static boolean isPrime(int n) {
// Corner case
if (n <= 1)
return false;
// Check from 2 to square root of n
for (int i = 2; i <= Math.sqrt(n); i++)
if (n % i == 0)
return false;
return true;
}
// Function to minimum operations required
// to reduce the number to a prime number
public static int minOperation(int N) {
// Because 1 cannot be converted to prime
if (N == 1)
return -1;
// If given number is already prime
// return 0
if (isPrime(N) == true) {
return 0;
}
// If number is not prime
else {
// Variable for total count
int count = 0;
int i = 2;
// If number is not equal to i
while (N != i) {
// If N is completely divisible by i
while (N % i == 0) {
// Temporary variable to store
// current number
int temp = N;
// Update the number by decrementing
// with highest divisor
N -= (temp / i);
// Increment count by 1
count++;
if (isPrime(N))
return count;
}
i++;
}
// Return the count
return count;
}
}
// Driver Code
public static void main(String args[]) {
int N = 38;
System.out.println(minOperation(N));
}
}
// This code is contributed by gfgking.
# python program for the above approach
import math
# Function check whether a number
# is prime or not
def isPrime(n):
# Corner case
if (n <= 1):
return False
# Check from 2 to square root of n
for i in range(2, int(math.sqrt(n)) + 1):
if (n % i == 0):
return False
return True
# Function to minimum operations required
# to reduce the number to a prime number
def minOperation(N):
# Because 1 cannot be converted to prime
if (N == 1):
return -1
# If given number is already prime
# return 0
if (isPrime(N) == True):
return 0
# If number is not prime
else:
# Variable for total count
count = 0
i = 2
# If number is not equal to i
while (N != i):
# If N is completely divisible by i
while (N % i == 0):
# Temporary variable to store
# current number
temp = N
# Update the number by decrementing
# with highest divisor
N -= (temp // i)
# Increment count by 1
count += 1
if (isPrime(N)):
return count
i += 1
# Return the count
return count
# Driver Code
if __name__ == "__main__":
N = 38
print(minOperation(N))
# This code is contributed by rakeshsahni
// C# program for the above approach
using System;
public class GFG {
// Function check whether a number
// is prime or not
public static bool isPrime(int n)
{
// Corner case
if (n <= 1)
return false;
// Check from 2 to square root of n
for (int i = 2; i <= Math.Sqrt(n); i++)
if (n % i == 0)
return false;
return true;
}
// Function to minimum operations required
// to reduce the number to a prime number
public static int minOperation(int N) {
// Because 1 cannot be converted to prime
if (N == 1)
return -1;
// If given number is already prime
// return 0
if (isPrime(N) == true) {
return 0;
}
// If number is not prime
else {
// Variable for total count
int count = 0;
int i = 2;
// If number is not equal to i
while (N != i) {
// If N is completely divisible by i
while (N % i == 0) {
// Temporary variable to store
// current number
int temp = N;
// Update the number by decrementing
// with highest divisor
N -= (temp / i);
// Increment count by 1
count++;
if (isPrime(N))
return count;
}
i++;
}
// Return the count
return count;
}
}
// Driver Code
public static void Main(string []args) {
int N = 38;
Console.WriteLine(minOperation(N));
}
}
// This code is contributed by AnkThon
<script>
// JavaScript program for the above approach
// Function check whether a number
// is prime or not
const isPrime = (n) => {
// Corner case
if (n <= 1)
return false;
// Check from 2 to square root of n
for (let i = 2; i <= parseInt(Math.sqrt(n)); i++)
if (n % i == 0)
return false;
return true;
}
// Function to minimum operations required
// to reduce the number to a prime number
const minOperation = (N) => {
// Because 1 cannot be converted to prime
if (N == 1)
return -1;
// If given number is already prime
// return 0
if (isPrime(N) == true) {
return 0;
}
// If number is not prime
else
{
// Variable for total count
let count = 0;
let i = 2;
// If number is not equal to i
while (N != i)
{
// If N is completely divisible by i
while (N % i == 0)
{
// Temporary variable to store
// current number
let temp = N;
// Update the number by decrementing
// with highest divisor
N -= parseInt(temp / i);
// Increment count by 1
count++;
if (isPrime(N))
return count;
}
i++;
}
// Return the count
return count;
}
}
// Driver Code
let N = 38;
document.write(minOperation(N));
// This code is contributed by rakeshsahni
</script>
Output
1
Time Complexity: O(sqrtN)
Auxiliary Space: O(1)