Given an integer **N**, the task is to find the number of factors of **N** which are perfect square.**Examples:**

Input:N = 100Output:4Explanation:

There are four factors of

100 (1, 4, 25, 100) that are perfect square.

Input:N = 900Output:8Explanation:

There are eight factors of 900 (1, 4, 9, 25, 36, 100, 225, 900) that are perfect square.

**Naive Approach:** The simplest approach to solve this problem is to find all possible factors of the given number **N** and for each factor, check if the factor is a perfect square or not. For every factor found to be so, increase **count**. Print the final **count**. **Time Complexity:** O(N) **Auxiliary Space:** O(1)

**Efficient Approach:**

The following observations need to be made to optimize the above approach:

The number of factors for a number is given by:

Factors of N = (1 + a

_{1})*(1 + a_{2})*(1 + a_{3})*..*(1 + a_{n})

where a_{1}, a_{2}, a_{3}, .., a_{n}are the count of distinct prime factors of N.

In a perfect square, the count of distinct prime factors must be divisible by 2. Therefore, the count of factors that are a perfect square is given by:

Factors of N that are perfect square = (1 + a

_{1}/2)*(1 + a_{2}/2)*…*(1 + a_{n}/2) where a_{1}, a_{2}, a_{3}, .., a_{n}are the count of distinct prime factors of N.

**Illustration:**

The prime factors of N = 100 are 2, 2, 5, 5.

Therefore, the number of factors that are perfect square are (1 + 2/2) * (1 + 2/2) = 4.

The factors are 1, 4, 25, 100.

Therefore, find the count of prime factors and apply the above formula to find the count of factors that are a perfect square.

Below is the implementation of the above approach:

## C++

`// C++ Program to implement ` `// the above approach ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function that returns the count of ` `// factors that are perfect squares ` `int` `noOfFactors(` `int` `N) ` `{ ` ` ` `if` `(N == 1) ` ` ` `return` `1; ` ` ` ` ` `// Stores the count of number ` ` ` `// of times a prime number ` ` ` `// divides N. ` ` ` `int` `count = 0; ` ` ` ` ` `// Stores the number of factors ` ` ` `// that are perfect square ` ` ` `int` `ans = 1; ` ` ` ` ` `// Count number of 2's ` ` ` `// that divides N ` ` ` `while` `(N % 2 == 0) { ` ` ` `count++; ` ` ` `N = N / 2; ` ` ` `} ` ` ` ` ` `// Calculate ans according ` ` ` `// to above formula ` ` ` `ans *= (count / 2 + 1); ` ` ` ` ` `// Check for all the possible ` ` ` `// numbers that can divide it ` ` ` `for` `(` `int` `i = 3; ` ` ` `i * i <= N; i = i + 2) { ` ` ` `count = 0; ` ` ` ` ` `// Check the number of ` ` ` `// times prime number ` ` ` `// i divides it ` ` ` `while` `(N % i == 0) { ` ` ` `count++; ` ` ` `N = N / i; ` ` ` `} ` ` ` ` ` `// Calculate ans according ` ` ` `// to above formula ` ` ` `ans *= (count / 2 + 1); ` ` ` `} ` ` ` ` ` `// Return final count ` ` ` `return` `ans; ` `} ` ` ` `// Driver Code ` `int` `main() ` `{ ` ` ` `int` `N = 100; ` ` ` ` ` `cout << noOfFactors(N); ` ` ` ` ` `return` `0; ` `} ` |

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## Java

`// Java program to implement ` `// the above approach ` `import` `java.util.*; ` ` ` `class` `GFG{ ` ` ` `// Function that returns the count of ` `// factors that are perfect squares ` `static` `int` `noOfFactors(` `int` `N) ` `{ ` ` ` `if` `(N == ` `1` `) ` ` ` `return` `1` `; ` ` ` ` ` `// Stores the count of number ` ` ` `// of times a prime number ` ` ` `// divides N. ` ` ` `int` `count = ` `0` `; ` ` ` ` ` `// Stores the number of factors ` ` ` `// that are perfect square ` ` ` `int` `ans = ` `1` `; ` ` ` ` ` `// Count number of 2's ` ` ` `// that divides N ` ` ` `while` `(N % ` `2` `== ` `0` `) ` ` ` `{ ` ` ` `count++; ` ` ` `N = N / ` `2` `; ` ` ` `} ` ` ` ` ` `// Calculate ans according ` ` ` `// to above formula ` ` ` `ans *= (count / ` `2` `+ ` `1` `); ` ` ` ` ` `// Check for all the possible ` ` ` `// numbers that can divide it ` ` ` `for` `(` `int` `i = ` `3` `; i * i <= N; i = i + ` `2` `) ` ` ` `{ ` ` ` `count = ` `0` `; ` ` ` ` ` `// Check the number of ` ` ` `// times prime number ` ` ` `// i divides it ` ` ` `while` `(N % i == ` `0` `) ` ` ` `{ ` ` ` `count++; ` ` ` `N = N / i; ` ` ` `} ` ` ` ` ` `// Calculate ans according ` ` ` `// to above formula ` ` ` `ans *= (count / ` `2` `+ ` `1` `); ` ` ` `} ` ` ` ` ` `// Return final count ` ` ` `return` `ans; ` `} ` ` ` `// Driver Code ` `public` `static` `void` `main(String[] args) ` `{ ` ` ` `int` `N = ` `100` `; ` ` ` ` ` `System.out.print(noOfFactors(N)); ` `} ` `} ` ` ` `// This code is contributed by 29AjayKumar` |

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## Python3

`# Python3 program to implement ` `# the above approach ` ` ` `# Function that returns the count of ` `# factors that are perfect squares ` `def` `noOfFactors(N): ` ` ` ` ` `if` `(N ` `=` `=` `1` `): ` ` ` `return` `1` ` ` ` ` `# Stores the count of number ` ` ` `# of times a prime number ` ` ` `# divides N. ` ` ` `count ` `=` `0` ` ` ` ` `# Stores the number of factors ` ` ` `# that are perfect square ` ` ` `ans ` `=` `1` ` ` ` ` `# Count number of 2's ` ` ` `# that divides N ` ` ` `while` `(N ` `%` `2` `=` `=` `0` `): ` ` ` `count ` `+` `=` `1` ` ` `N ` `=` `N ` `/` `/` `2` ` ` ` ` `# Calculate ans according ` ` ` `# to above formula ` ` ` `ans ` `*` `=` `(count ` `/` `/` `2` `+` `1` `) ` ` ` ` ` `# Check for all the possible ` ` ` `# numbers that can divide it ` ` ` `i ` `=` `3` ` ` `while` `i ` `*` `i <` `=` `N: ` ` ` `count ` `=` `0` ` ` ` ` `# Check the number of ` ` ` `# times prime number ` ` ` `# i divides it ` ` ` `while` `(N ` `%` `i ` `=` `=` `0` `): ` ` ` `count ` `+` `=` `1` ` ` `N ` `=` `N ` `/` `/` `i ` ` ` ` ` `# Calculate ans according ` ` ` `# to above formula ` ` ` `ans ` `*` `=` `(count ` `/` `/` `2` `+` `1` `) ` ` ` `i ` `+` `=` `2` ` ` ` ` `# Return final count ` ` ` `return` `ans ` ` ` `# Driver Code ` `if` `__name__ ` `=` `=` `"__main__"` `: ` ` ` ` ` `N ` `=` `100` ` ` ` ` `print` `(noOfFactors(N)) ` ` ` `# This code is contributed by chitranayal ` |

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## C#

`// C# program to implement ` `// the above approach ` `using` `System; ` ` ` `class` `GFG{ ` ` ` `// Function that returns the count of ` `// factors that are perfect squares ` `static` `int` `noOfFactors(` `int` `N) ` `{ ` ` ` `if` `(N == 1) ` ` ` `return` `1; ` ` ` ` ` `// Stores the count of number ` ` ` `// of times a prime number ` ` ` `// divides N. ` ` ` `int` `count = 0; ` ` ` ` ` `// Stores the number of factors ` ` ` `// that are perfect square ` ` ` `int` `ans = 1; ` ` ` ` ` `// Count number of 2's ` ` ` `// that divides N ` ` ` `while` `(N % 2 == 0) ` ` ` `{ ` ` ` `count++; ` ` ` `N = N / 2; ` ` ` `} ` ` ` ` ` `// Calculate ans according ` ` ` `// to above formula ` ` ` `ans *= (count / 2 + 1); ` ` ` ` ` `// Check for all the possible ` ` ` `// numbers that can divide it ` ` ` `for` `(` `int` `i = 3; i * i <= N; i = i + 2) ` ` ` `{ ` ` ` `count = 0; ` ` ` ` ` `// Check the number of ` ` ` `// times prime number ` ` ` `// i divides it ` ` ` `while` `(N % i == 0) ` ` ` `{ ` ` ` `count++; ` ` ` `N = N / i; ` ` ` `} ` ` ` ` ` `// Calculate ans according ` ` ` `// to above formula ` ` ` `ans *= (count / 2 + 1); ` ` ` `} ` ` ` ` ` `// Return final count ` ` ` `return` `ans; ` `} ` ` ` `// Driver Code ` `public` `static` `void` `Main(String[] args) ` `{ ` ` ` `int` `N = 100; ` ` ` ` ` `Console.Write(noOfFactors(N)); ` `} ` `} ` ` ` `// This code is contributed by PrinciRaj1992` |

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**Output:**

4

**Time Complexity:** O(log(N)) **Space Complexity: **O(1)

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