# Find the smallest number X such that X! contains at least Y trailing zeros.

Given an integer Y, find the smallest number X such that X! contains at least Y trailing zeros.

Prerequisites – Count trailing zeroes in factorial of a number

**Examples:**

Input :Y = 2

Output :10

10! = 3628800, which has 2 trailing zeros. 9! = 362880, which has 1 trailing zero. Hence, 10 is the correct answer.

Input :Y = 6

Output :25

25! = 15511210043330985984000000, which has 6 trailing zeros. 24! = 620448401733239439360000, which has 4 trailing zeros. Hence, 25 is the correct answer.

**Approach:** The problem can be easily solved by using **Binary Search**. The number of trailing zeros in N! is given by the count of the factors 5 in N!.Read this article for prerequisites. The countFactor(5, N) function returns the count of factor 5 in N! which is equal to count of trailing zeros in N!. The smallest number X such that X! contains at least Y trailing zeros can be computed quickly by using binary search on a range [0, 5 * Y] using this function.

Below is the implementation of above approach.

## C++

`// C++ implementation of the approach ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function to count the number ` `// of factors P in X! ` `int` `countFactor(` `int` `P, ` `int` `X) ` `{ ` ` ` `if` `(X < P) ` ` ` `return` `0; ` ` ` ` ` `return` `(X / P + countFactor(P, X / P)); ` `} ` ` ` `// Function to find the smallest X such ` `// that X! contains Y trailing zeros ` `int` `findSmallestX(` `int` `Y) ` `{ ` ` ` `int` `low = 0, high = 5 * Y; ` ` ` `int` `N = 0; ` ` ` `while` `(low <= high) { ` ` ` `int` `mid = (high + low) / 2; ` ` ` ` ` `if` `(countFactor(5, mid) < Y) { ` ` ` `low = mid + 1; ` ` ` `} ` ` ` ` ` `else` `{ ` ` ` `N = mid; ` ` ` `high = mid - 1; ` ` ` `} ` ` ` `} ` ` ` ` ` `return` `N; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `int` `Y = 10; ` ` ` ` ` `cout << findSmallestX(Y); ` ` ` ` ` `return` `0; ` `} ` |

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## Java

`// Java implementation of the above approach ` `class` `GFG ` `{ ` ` ` ` ` `// Function to count the number ` ` ` `// of factors P in X! ` ` ` `static` `int` `countFactor(` `int` `P, ` `int` `X) ` ` ` `{ ` ` ` `if` `(X < P) ` ` ` `return` `0` `; ` ` ` ` ` `return` `(X / P + countFactor(P, X / P)); ` ` ` `} ` ` ` ` ` `// Function to find the smallest X such ` ` ` `// that X! contains Y trailing zeros ` ` ` `static` `int` `findSmallestX(` `int` `Y) ` ` ` `{ ` ` ` `int` `low = ` `0` `, high = ` `5` `* Y; ` ` ` `int` `N = ` `0` `; ` ` ` `while` `(low <= high) ` ` ` `{ ` ` ` `int` `mid = (high + low) / ` `2` `; ` ` ` ` ` `if` `(countFactor(` `5` `, mid) < Y) ` ` ` `{ ` ` ` `low = mid + ` `1` `; ` ` ` `} ` ` ` ` ` `else` ` ` `{ ` ` ` `N = mid; ` ` ` `high = mid - ` `1` `; ` ` ` `} ` ` ` `} ` ` ` ` ` `return` `N; ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `main(String args[]) ` ` ` `{ ` ` ` `int` `Y = ` `10` `; ` ` ` ` ` `System.out.println(findSmallestX(Y)); ` ` ` `} ` `} ` ` ` `// This code is contributed by Ryuga ` |

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## Python3

`# Python3 implementation of the approach ` ` ` `# Function to count the number ` `# of factors P in X! ` `def` `countFactor(P, X): ` ` ` `if` `(X < P): ` ` ` `return` `0` `; ` ` ` ` ` `return` `(X ` `/` `/` `P ` `+` `countFactor(P, X ` `/` `/` `P)); ` ` ` `# Function to find the smallest X such ` `# that X! contains Y trailing zeros ` `def` `findSmallestX(Y): ` ` ` ` ` `low ` `=` `0` `; ` ` ` `high ` `=` `5` `*` `Y; ` ` ` `N ` `=` `0` `; ` ` ` `while` `(low <` `=` `high): ` ` ` `mid ` `=` `(high ` `+` `low) ` `/` `/` `2` `; ` ` ` ` ` `if` `(countFactor(` `5` `, mid) < Y): ` ` ` `low ` `=` `mid ` `+` `1` `; ` ` ` ` ` `else` `: ` ` ` `N ` `=` `mid; ` ` ` `high ` `=` `mid ` `-` `1` `; ` ` ` ` ` `return` `N; ` ` ` `# Driver code ` `Y ` `=` `10` `; ` ` ` `print` `(findSmallestX(Y)); ` ` ` `# This code is contributed by mits ` |

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## C#

`// C# implementation of the approach ` `class` `GFG ` `{ ` ` ` `// Function to count the number ` `// of factors P in X! ` `static` `int` `countFactor(` `int` `P, ` `int` `X) ` `{ ` ` ` `if` `(X < P) ` ` ` `return` `0; ` ` ` ` ` `return` `(X / P + countFactor(P, X / P)); ` `} ` ` ` `// Function to find the smallest X such ` `// that X! contains Y trailing zeros ` `static` `int` `findSmallestX(` `int` `Y) ` `{ ` ` ` `int` `low = 0, high = 5 * Y; ` ` ` `int` `N = 0; ` ` ` `while` `(low <= high) ` ` ` `{ ` ` ` `int` `mid = (high + low) / 2; ` ` ` ` ` `if` `(countFactor(5, mid) < Y) ` ` ` `{ ` ` ` `low = mid + 1; ` ` ` `} ` ` ` ` ` `else` ` ` `{ ` ` ` `N = mid; ` ` ` `high = mid - 1; ` ` ` `} ` ` ` `} ` ` ` ` ` `return` `N; ` `} ` ` ` `// Driver code ` `static` `void` `Main() ` `{ ` ` ` `int` `Y = 10; ` ` ` ` ` `System.Console.WriteLine(findSmallestX(Y)); ` `} ` `} ` ` ` `// This code is contributed by mits ` |

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## PHP

**Output:**

45

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